It's my pleasure to speak with you this afternoon about competition and monopoly. Super interesting and timely topic. Now, for several years at Mises University, the lecture on competition and monopoly and antitrust and regulation and so forth was given by my dear friend, Walter Block. And Walter came of age in the 60s. And it was sort of a different era back then in macroeconomics. The Keynesian model was dominant. And even the Chicago School was kind of considered a fringe movement. In microeconomics, everyone was talking about John Kenneth Galbraith and his sort of critiques of big companies. And antitrust enforcement was a really big deal. People were talking about the robber barons all the time. And so Walter, when he spoke about competition and monopoly, even until recently, he would give a lot of examples about the robber baron era. And he'd talk about Standard Oil and so forth. I mean, I think he knew those cases. Well, Walter, I think he was an expert witness in the Standard Oil case of 1911. But who cares about that stuff today, right? Now we're all about markets that are really dynamic and all economists, even mainstream economists, are in favor of robust competition. And they don't worry about monopoly so much. And all that sort of antitrust body of antitrust experience is really more relevant to understanding the past than understanding the present. But oh no, such is not the case. You see increasing complaints nowadays about the big tech companies being the standard oils of today, Tim Cook and Zuckerberg and Jeff Bezos. These are the robber barons of our day. And they're much worse and more nefarious than John D. Rockefeller and Andrew Carnegie and all these other folks that your high school history teacher taught you were the great villains of modern times. And in fact, the US has not yet brought a major antitrust suit against Google or Apple or Facebook or Amazon for attempts to monopolize their respective markets. But it would not surprise me in the least if such a case were on the horizon. Some of you may recall or have read about 20 years ago, 25 years ago, when Microsoft was considered to be the dominant player in tech, a company that could never be pushed from its perch as the most important IT firm because of its dominance of the desktop computer or the operating system running the desktop computer. Those of you millennials in the crowd might be thinking, what's a desktop computer? So Microsoft obviously doesn't wield the clout that it used to. And who knows what the future holds for these guys. But of course, the government is very active in playing this role in sort of steering markets, especially in technology markets. One of the biggest cases in recent months was an attempt by the US Justice Department, the Trump Justice Department, to block the merger of AT&T and Time Warner, and the judge ruled against the Justice Department in favor of allowing the merger to go through. But notice the headline here. Judge clears $85 billion bid for Time Warner with no conditions, like how shocking, right? We live in an era in which when large companies want to do things like perhaps combine their assets with those of other large companies, they have to get permission, de facto, right? And a judge with the assistance of regulators and bureaucrats, right? And other experts will make a decision about what we will allow this company to do. Well, no, we're not gonna let you merge. Well, okay, we'll allow it, but you gotta do this and this and this, right? That's kind of the norm for today. And my point is the fact that we live in the internet age and that many people, even mainstream economists, talk about dynamic competition and looking at processes through time and making sure that innovation plays its proper role. By no means does that suggest that antitrust enforcement is weaker than it was before or that mainstream economists are less worried about so-called monopoly problems than they were. So let's talk about competition and monopoly from an Austrian perspective. So what do I mean by competition? That's sort of a very commonplace word, right? We talk about competing all the time and everyday language, usually what we mean by competition is one of two things, right? Either there's just a word for kind of a contest, a situation in which individuals or groups are trying to do better than some other individuals or groups at performing some task. It's a big competition at school or in a race who can run the fastest race or which team can win the World Cup, right? That's a competitive event. I like competitive cooking myself. That's one of my hobbies to watch on TV, maybe to participate someday if I practice hard. We also use the word in a more descriptive sense of a situation like that that is particularly intense, like the written and oral exams that means as you, we say they are highly competitive. What we mean is it's hard, but I don't wanna discourage any of you from participating, of course. But everybody knows what it means to describe something as competitive. Now, what about kind of legal and economic notions? Well, in a more kind of technical sense, right? Historically, the legal meaning of the term competition or the terms competition and monopoly has referred to sort of legal restrictions or legal barriers to entry. So in kind of the common law notion, a market or an industry is competitive if the government does not interfere with it, if the government takes a hands-off posture and lets anyone who, any merchant, any producer, any industrialist who wants to give this industry a try, who wants to enter this market can go for it and neither succeed or not succeed, right? Monopoly referred to the condition of government interference in the market, right? For example, by the state granting an exclusive right for a particular entrepreneur or firm to be in some industry. The East India Tea Company had a legal monopoly from the British Crown, of course, the Dutch East India Company had a similar monopoly from the Dutch Crown and their government said, no other Dutch ship or no other English ship can legally sail out to the Spice Islands and bring back tea and spices and so forth, right? That's what people understood by monopoly when the government intervenes in a market and favors one particular entrepreneur or firm over another. But those of you who are economic students or if you keep up with any of the contemporary literature and jargon know that that is not at all what competition means to most economists, right? In kind of mainstream economics and neoclassical economics, you know, competition, terms like competition and monopoly refer to particular abstract, artificial constructs or models, in particular, the concepts of so-called perfect competition or imperfect competition, which I'll describe in more detail in just a moment. And in mainstream economics, if any firm has the ability to charge prices higher than the actual marginal cost of production, i.e. to earn a profit, then that firm must not be operating under competitive conditions, right? Monopoly is defined as the ability to increase price over marginal cost to charge prices higher than cost, which, I mean, so basically any profitable firm has some kind of monopoly privilege, according to sort of standard, the standard approach. Everybody's talking about privilege nowadays, profit privilege, right? If you're in a profit, you got monopoly privilege of some kind. Now the Austrians have, as it won't surprise you to know, have a very different perspective on monopoly and the meaning of competition, though there's some variation within the Austrian tradition, as I'll suggest in just a moment. Okay, so just a brief review of the neoclassical approach before we go into some criticism. This will be familiar to those of you who are economics majors. What mainstream economists mean when they talk about perfect competition or imperfect competition or various flavors of competition is really a way of characterizing the structural features of various markets, right? So we look at a particular market theoretically or we look at an actual market in the world and we say, well, okay, so, you know, is it a bunch of producers selling kind of an identical product, a homogeneous product or is each firm selling a differentiated product? You know, is each product, are certain products completely unique such that there are no substitutes for them whatsoever? First we gotta assess that, right? Then we have to look at how many buyers and sellers. Are there lots of buyers and sellers or just a few? Are they kind of equal in their spending power or their buying power or selling power or some of them much larger or some of them much smaller? How difficult is it to enter a new market or exit an existing market? Not just in the legal sense that I talked about just before, but even, you know, in an economic sense, right? I mean, it's just hard, right? So, you know, if I wanna be a successful, you know, pizza entrepreneur, I gotta have some knowledge of how to make pizza, right? I've gotta be able to get the funds to set up a, you know, to rent a store and to get a pizza oven and so forth. And if I don't have enough money to do that, well, then entry is restricted, right? That's a barrier to entry that might be a concern from an antitrust perspective, right? Anything that impedes my ability to enter any business that I can dream of costlessly is potentially an entry barrier, right? So, essentially, every market is characterized by entry barriers in this sense. No market is really competitive and therefore it is appropriate, potentially, for the government to intervene in any market kind of situation. And, you know, you might look at information conditions if there's any private information or hidden information that could cause a problem as well. So, the wheat market, right? So, every economics textbooks, favorite example of a perfectly competitive industry, which it really isn't, of course, in which there is a very large number of sellers, lots of little wheat farmers, and it's all just wheat.  As opposed to a market-like large-body commercial aviation in which, for passenger planes, there are only two firms, right, in the market today, Airbus and Boeing. Of course, there have been other firms in the past, McDonald Douglas and Lockheed and so forth, but basically there are two firms in the business of selling large-body commercial jets and major economists call that a duopoly, right? It's a market with just two firms. Or, there may be cases where there's only one firm in the industry. If we define the industry as the search engine that everybody uses, right, then Google has a monopoly. Well, I mean, yeah, technically there's like Bing, right, but when have you ever Binged something to get some information, right? So, the point is, economists wanna look at these, you know, different markets and say, okay, our first task is to say, well, is this market, is it close to the sort of theoretical notion of perfect competition? Is it close to pure monopoly? Is it more like a duopoly or an oligopoly with three or four or five firms? And then we decide what to do about it. Of course, as I mentioned, none of these cases strictly fit the definitions, right? I mean, if you wanna be literal about it, the definition of perfect competition includes, you know, it requires that a market have an infinite number of firms, each of which in the limit has a size, you know, assets or sales of zero. I haven't seen any of those, right? But most economists get lazy. They say, well, if there's lots and lots of little firms, that's good enough. Okay, of course, even wheat is not really a homogeneous product because you got lots of different kinds of wheat. You've got number three wheat and number four wheat and winter wheat and red wheat and you've got nowadays, you've got organic wheat and GMO free wheat and all different kinds of wheat. But to none of these examples completely fit. I mean, there are substitutes for large body commercial airplanes, namely the smaller ones. They're made by firms like Embraer and Canadair and so forth. And of course, there are substitutes for all of the products that Google offers. We just don't think they're as good. Okay, that's not, this would be the literal definition of monopoly. So let's look at, you know, two examples, the perfectly competitive firm first and then a firm that has a monopoly, right? How many of you have seen this kind of diagram before? Right, lots of you. So imagine, this is a neoclassical economics treatment. Imagine a market like the wheat market. We assume that there are lots and lots and lots of little tiny wheat farmers, each of whom supplies wheat to the wheat market.  It just all goes into the wheat market. Therefore, from the perspective of an individual farmer, the story goes, that farmer, you know, can sort of bring as much wheat to market as he or she wants without affecting the market price. Right, because each farmer is so small relative to the market, you know, you could dump all your available wheat on the market. That's not gonna drive the price down because you're so small relative to the whole market. Likewise, you know, you could withhold wheat from the market.  So in that sense, the demand curve facing the individual firm is perfectly elastic. That's a horizontal demand curve, meaning the market price doesn't change no matter what quantity an individual firm brings to market. Of course, if all firms increase their wheat production or all firms decrease their wheat production in tandem, in harmony, right, then the market price would be driven up or down. But if each farmer assumes that other farmers are doing their own thing, then the farmer can supply as little or as much as he or she wants without affecting the price. That's why you get that horizontal demand curve. And of course, marginal revenue is just equal to price in this case. The amount of revenue you get from selling one bushel of wheat is always the same, right? We could talk about how that affects the ability to calculate marginal revenue product like we were doing yesterday, but we won't do that now. So imagine that this individual firm has an upward-sloping marginal cost curve, you know, U-shaped average cost curve, then in kind of a long-run equilibrium, right, each firm will be producing the quantity of wheat at which the price is just equal to the marginal cost. And if there's sort of entry and exit, the ability for different firms to enter and exit depending on market conditions, you kind of settle in this place where you get this long-run competitive equilibrium in which that particular price quantity pair right here, right, is also at the minimum of the average cost of production. So you get two things here that are supposed to be, you know, desirable from an efficiency standpoint, right? All the time, both in the short run and the long run, firms are charging prices equal to marginal costs. So when consumers are deciding how much wheat they wanna buy or bakers are deciding how much wheat they wanna buy as an input, right? They're making that choice using a number, the price that also corresponds to sort of the opportunity cost to society of using a bushel of wheat. And therefore you get this sort of perfect allocative efficiency. And in the long run, because if this price marginal cost point is above the average cost of production, firms will earn profits, other firms will enter, that'll drive down the market price. And if the price marginal cost pair lies below the average total cost curve, then firms will leave the industry and that will bring the price up. You get this kind of equilibration where in the long run, right, the price that each farmer charges is also exactly equal to the minimum of the average cost of production. It's like this is the cheapest possible way we could get wheat. Okay, so everybody's happy except farmers, right? Because farmers don't earn any economic profit, right? Each farmer earns just enough to compensate him for not becoming a pizza entrepreneur or whatever his next best opportunity is. Okay, there's nothing left over, there's no residual, there's no economic profits. So your economics professor, unless you're lucky enough to have an Austrian professor, will say, look everybody, this is what a market ought to be from an individual producer's point of view. When we have a world like this, everything's great, everything's efficient, unfortunately, because of these robber barons and other bad people, some markets are not like this. So we've got to do something to fix them to make them more like this, right? In many cases, your professor tells you firms do not operate in a perfectly competitive environment, they hold something called market power, right? So a firm with market power is in an entirely different situation. What kind of a firm is this? Well, market power exists and sort of is defined by the condition where the firm faces a downward sloping demand curve for its product, okay? I want people to buy more of my pizza, I've got to charge a lower price per pizza, right? I mean, we assume that demand curves are downward sloping in markets. So if the demand curve from my product, from my point of view is also downward sloping, then I no longer face this situation with a perfectly elastic horizontal demand curve, which allows me to do something very sneaky, right? So if you, as you know, those of you studying this, when the demand curve is downward sloping, right, that's price as a function of quantity, then the amount received on the margin from selling one more unit, i.e. the marginal revenue is not exactly in the same place. The marginal revenue curve does not coincide with the demand curve, right? It's more steeply sloped than the demand curve and it lies below the demand curve. So that green line represents the marginal revenue from selling one, two, three, four, five units. The point is marginal revenue decreases more quickly than price. But because, you know, to sell more units on the market, you have to lower the price, not just for the last unit sold, but for all of the units that you sell, all the ones that came before it. So on the margin, you're actually adding even less to your revenue than the price. Who cares, what's the point? Well, if you have a marginal revenue curve that is below the demand curve and more steeply sloped, when the profit maximizing firm, profit maximizing entrepreneurs says, well, I wanna produce the quantity where marginal revenue is equal to marginal cost. Notice where that is, that's right here, right? Call that QM, the monopoly quantity, right? So the firm maximizes profits where marginal revenue, that's the green curve, is equal to marginal cost, which is right here. What price does the firm charge at that quantity? Well, the highest price that you can get and still be able to sell that much stuff, which is not right over here. In fact, you sort of march all the way up to the demand curve and scoot over to the vertical axis. So the monopoly price is right here, right? So notice there's this kind of wedge, right? The price that the monopolist charges when the monopolist is maximizing profits is greater than the marginal cost of production at that quantity, okay? So price is higher than marginal cost, which means what? Left over money for that greedy monopolist, okay? So what you get in this sort of standard analysis, right? That because the firm with market power faces a downward sloping demand curve, that firm maximizes profits by producing the quantity at which marginal revenue equals marginal cost, same as any firm. But at that place in this situation, the price is greater than the marginal cost. And the way it sometimes described is the following. If this market were perfectly competitive, right? And we're looking at the whole industry, not just an individual firm, a perfectly competitive industry would have an equilibrium price quantity combination here, right, where the marginal cost curve hits the demand curve. So this is how much output would be produced if this were a perfectly competitive market with lots of firms. But because it's only one firm, the firm decides to produce less. The firm deliberately restricts quantity relative to the perfectly competitive amount in order to ride up that demand curve and jack up the price from PC to PM. Thus earning profits, thus lining the monopolist's pockets with ill-gotten gains, okay? That's the standard story. They don't usually make it as dramatic as that. That's a bad outcome, right? Because it means profits for the firm and less well-being or surplus for consumers. And there's also this little yellow piece, the so-called dead-weight loss, which nobody gets subject for another day. But the point is in sort of this kind of standard approach, you start by describing a kind of a market that could never exist. Define that as the ideal from a welfare or efficiency point of view. And then use that to show that all actually existing markets are inefficient, do not maximize social welfare. And therefore there's some case to be made for fixing this somehow. What can we do to make this market more like the previous one? Well, we could say, oh, there's just one firm in the market, that's bad. Let's make it two or five or 10 or 100 firms. How do we do that? Well, we file an antitrust suit and force the firm to be broken up into little tiny firms. Or maybe we just leave the firm as it is, but we regulate the price. We allow you to exist as one firm. But we command that you charge this price instead of this price, public utilities fit in that example of that kind of scenario. Now, I'm being a little bit, I guess not exactly facetious, but I'm kind of overstating the case. Most mainstream economists, most neoclassical economists, if you press them on these points, we'll say, well, yeah, it's really not that simple. In fact, there are some problems with this approach to monopoly, the standard approach to monopoly. For example, one of you famously associated with Joseph Schumpeter, Schumpeter over here with the flower, who's kind of a dandy. Schumpeter said, look, yeah, I mean, profits are, monopoly is bad in the standard way. It means more money for the firm and less well-being for the consumer. And you can have dead weight losses, but that's kind of the price we have to pay as a society to get innovation, because Schumpeter thought that innovations primarily come from large firms and large firms need a budget. They need an R&D budget to be able to do experiments and improve products and so forth. So if in a perfectly competitive world, we would never get any improvement because firms don't have any money left over for research. So we have to allow some monopoly because that gives us more economic growth in the long run. You know, other sort of sophisticated thinkers like Harold Demsets, who I would describe as a kind of Chicago style neoclassical economist, but a very thoughtful one, says, look, you know, how do you know that monopoly exists? Well, simply looking at market shares as people often do is not a good indicator, right? And of course, we all know this. It's not the case that Google, somehow mysteriously dominates the search market and uses that to screw us over somehow. Or I'm still waiting to find out how they're screwing us over by offering us free search results and maps and mail and all the other stuff they offer.  Well, because they're better. I mean, they offer products and services that people prefer to other products and services. And you know, through learning by doing and economies of scale and so forth, they get pretty good at it. Yeah, it's hard for someone else to compete with Google in the search business, but it's not because Google did anything inappropriate. Google just offered a better product than those of its rivals, okay? So why would we wanna penalize firms for being good at what they do? Because being good at what you do tends to lead to growth.  What else are you supposed to do? Just quit and go home. There's also the very interesting literature that from the late 70s, early 80s, by people like William Baumall, late William Baumall on what they call contestable markets or contestability. The idea that even in markets that are not, even markets that are not perfectly competitive that have just a handful of firms in them, may still look a lot like competitive industries in the standard sense if there is the potential for other firms to enter, right? So even if I'm the only pizza vendor in town, I'm not gonna jack my prices way up above marginal costs because if I did so, then somebody else would just set up a pizza shop across the street. It's not very hard to do. And then I would lose money, I don't wanna deal with that. So just the threat of potentially having to compete with some rival might discipline my behavior as an incumbent, even if that rival is not currently in the market. Okay, so what can we say about this from sort of an Austrian point of view? Well, let's start kind of with sort of first principles, right? Remember, Austrian economics is not, Austrians don't attempt to describe phenomena using abstract mathematical models, but rather try to trace out cause and effect in a realistic fashion, hence causal realist economics, looking at the actual choices made by real persons and theorizing about real human action. And of course, real human action does not exist in terms of infinite quantities and infinitely small firms and perfectly smooth and continuous curves, right? So think about this way. I mean, each seller, no matter how small, contributes a discrete quantity to the market. So, I mean, conceptually there doesn't make sense to say that for any demand curve really to be perfectly elastic. I mean, you know what that means mathematically, a perfectly straight line, right? I mean, even a very small wheat farmer is adding some quantity to the wheat market. Now, you might say, well, it's only, you know, 100 bushels and the wheat market has millions and millions of bushels. Okay, well then that action would have very small effect on price. And depending on how prices are quoted, right? If they're quoted in certain increments, that may not be enough of an increase in quantity to bump the price needle, you know, down to the next lowest increment. But it doesn't follow, that's not the same thing as saying that the firm faces a perfectly elastic demand curve. There's no infinitesimally small units. So basically all firms, no matter how large or small, no matter how many competitors they have, face the same kind of situation, right? We discussed this in my entrepreneurship talk yesterday. All entrepreneurs are trying to earn profits, right? By seeking combinations of resources that will yield what they expect to be revenues greater than their costs. They're doing so under uncertainty, right? Whether you're the only firm currently offering a particular product or whether there are many firms offering that same product, whether there are lots of entrepreneurs trying to get those same inputs or just a few, from a subjective perspective, the problem facing the entrepreneur is exactly the same, right? There's no actual knowledge and technical differences and so forth, but from a praxeological point of view, there's no difference between a competitive firm and a monopolized firm in terms of the objective of the entrepreneur and the means the entrepreneur uses to try to achieve that objective. But we can think about some more specific issues. I mean, what is the following? Requiring firms to increase their output beyond what the entrepreneur considers to be the profit maximizing quantity, right? The entrepreneur's desired quantity. In a sense, it's a violation of the entrepreneur owner's rights. Okay, this is easy to see with certain kinds of examples. Apparently from what I read in 2017, the highest paid actor, the highest paid Hollywood celebrity is Mark Wahlberg. I was surprised, I thought it would be Tom Cruise maybe, but Tom Cruise is like 15th, okay? So Mark Wahlberg is currently the highest paid actor in Hollywood. He's in a lot of movies, right? Some of them are good, some of them are not so good. But you know, you can say, well, look, you know, there's only one Mark Wahlberg. I mean, there are some substitutes, but they're not perfect substitutes, right? So if consumers really desire to see Mark Wahlberg in movies, right, and would be willing to pay enough at the box office so that Mark Wahlberg could pocket enough earnings from doing movies to make it worth his while, then I mean, basically he should be working every minute of every day, right? Let's say Mark Wahlberg makes 10 feature films per year. Well, he should make 20 or 50 or 100, right? That might increase consumer welfare because we'd be willing to pay enough on the margin to make him at least break even from doing all those additional movies.  I imagine, I don't know the guy personally, but if he's like other Hollywood stars, when he's not currently filming a movie or doing other crazy stuff that those people do, you know, he's looking at scripts, right? His agent gets scripts and they make pitches to Mark Wahlberg and they send it to him. You know, would you be interested in doing this film? No, we'd love for you to play this part. I mean, he's selective. He chooses what parts he wants to play because if he's in every single movie and every single TV show and every single commercial, then we're gonna get, you know, Wahlberg fatigue and that's probably gonna lower his value on the market. All right, so each of us makes, we make ourselves our labor available on the market in the way that we see is appropriate. And it wouldn't make sense to say, oh, you have to work more hours because consumers want to see you more. Okay, well, sorry.  I don't want to. And it would be weird to sort of force me to give another lecture. Okay, well, if it applies to Mark Wahlberg who has a monopoly on Mark Wahlberg or Peter Klein who has a monopoly on Peter Klein lectures, I mean, how's it really any different with Jeff Bezos? Right, except that he has 150 billion reasons to do whatever the heck he wants. You know, why should the owners of Amazon, let's say that Amazon is one of these, is a monopoly and Jeff Bezos is a robber baron and blah, blah, blah, right? So essentially what the implication is, Amazon is not supplying enough product to the market. Right, Amazon is not producing as much as it conceivably could so that it can charge higher prices than it would charge under more competitive conditions. Right, or standard oil or anybody. If the argument is the monopolist restricts quantity in order to be able to charge a higher price and the remedy is to require the monopolist to increase quantity, well then, I mean, then everybody should be required to offer whatever services consumers want, right? But we recognize that that would be a violation of individual sovereignty and that would reduce welfare in the sort of meaningful Austrian sense of that word. There's a lot of other problems too. I mean, where does elasticity of demand come from?  Or remember, elasticity is not something given by nature and elasticity is not something that the entrepreneur sort of creates. No, elasticity reflects people's willingness to pay certain prices for certain quantities of the good. Elasticity comes from the consumer, right? Elasticity of demand reflects consumer preferences. So if you say, well, Google has too much market power because none of the other search engines are as good and therefore Google sort of forces people to use Google search and therefore it can force you to watch a bunch of ads, see a bunch of ads you don't really wanna see and it can steal all your information, et cetera. I'm gonna talk about that later in the week. Lecture on the economics of data privacy. But I mean, look, nobody forces you to use Google. Nobody forces you to use Amazon or Netflix or whatever. Nobody forced anyone to purchase the products of Standard Oil or Carnegie Steel, right? If in fact those firms face an inelastic demand for their product or a less than perfectly elastic demand, thus allowing them to earn monopoly profits, what of it? Right? That's a reflection of subjective consumer preferences. We choose, you know, you say, oh, well, there's no good substitute for Google. Well, that's because that's in our minds. We have decided we like Google better. It's not like some engineering thing because yeah, I mean, Bing is kind of the same thing as Google. We just don't like it as much, okay? So what makes one product a substitute for another product and therefore the number of substitutes available for a given product in determining whether or not it has a monopoly, quote unquote, exists in the minds of consumers. It's not given by engineers or given by nature, okay? So what about monopoly in the common law meaning, right? So as you may know, in Murray Rothbard's writing, he adopted this kind of common law notion of monopoly and said, look, if we go back to that common law definition, we can analyze monopoly in terms of government privilege, right, a monopoly, a firm has monopoly power if the government somehow restricts other firms or other entrepreneurs from competing with the favored entrepreneur, the favored firm. The judge last night talked about the Lysander Spooner post office case. I don't know if you guys knew about that case, but Spooner argued, I think it was about 1844. He argued that, well, the Constitution does give the federal government in the U.S. the authority to operate a post office, but it does not give the post office a monopoly. There's nothing in the Constitution about limiting competition to the post office. So he opened up his own post office and he charged rates that were much lower than the U.S. Postal Service rates. And so eventually they went after him and Congress passed a law in response to Lysander Spooner making it illegal for anybody to deliver first class mail. That's not in the Constitution. That's congressional legislation. So the U.S. Postal Service and the post office in most countries has a monopoly in the common law sense because it's illegal for anybody else to compete with it. A lot of local electric companies, water companies and so forth have exclusive rights to produce that good or service. A patent, even those who defend patents and that's a controversial area, can see that a patent is a kind of monopoly. A patent is a temporary legal right to be the exclusive producer of a certain thing or exclusive user of a certain technology. Now, people who defend patents argue kind of a la schumpeter that well, yeah, patents are not good from a sort of market efficiency point of view but we have to give patents, we have to give short term, 17 year temporary monopolies to induce people to invent new stuff. Tariffs and quotas can be sources of monopoly privilege as well, right? I mean, it's currently not illegal for Chinese steel makers to sell their steel in the United States but Trump has made it a lot more costly than to do so. Trump has talked about banning, he doesn't like the fact that if you drive up down Fifth Avenue in New York, you see a lot of Mercedes Benz limousines. He says those should be Cadillacs or Lincoln Continentals and he wants to ban German automakers from exporting to the US although it's not even clear what that means since most Mercedes sold in the US are made in Alabama just up the road from here. But if there were some kind of a tariff on, if all BMWs were made in Germany and there's a 50% tariff, that's kind of like monopoly protection for American auto producers, right? Now there are some less obvious sources of monopoly privilege, certain kinds of regulation and even tax policy. So you could argue that progressive taxation, right? Which limits capital accumulation is a kind of, provides a kind of privilege for people who already have a lot of capital, right? It's hard for me to amass a fortune to compete with Jeff Bezos or whatever. He's already got a fortune, right? So he might be in favor of high progressive taxation because it makes it harder for somebody else to be the next Jeff Bezos, okay? A lot of environmental and labor rules, right? They say, well, we just want to protect workers, we want to make sure that firms protect the environment and so we're gonna impose costly regulation on firms.  You can make sure that all of your facilities are accessible and so forth.  In fact, that may make it prohibitively costly for a mom and pop startup to compete with a dominant incumbent. So what about regulation, right? Including antitrust policy here is sort of a form of regulation. Well, one approach to, in the standard literature, is to say, well, if we can't break up the monopolist and transform the industry into a competitive one, we can at least regulate the monopolist and sort of force the monopolist to produce something like the competitive price and quantity. Well, you have a knowledge problem there. I mean, what is the perfectly competitive price and quantity? All we know from a praxeological point of view is that real prices that exist in real markets are those that emerge from a competitive in our sense process in which entrepreneurs can try different things and so forth. We don't know what these hypothetical, perfectly competitive prices are. And of course, the idea of setting up a government agency with the authority to regulate price and quantity for firms that opens up a whole nother Pandora's box. Right now you've got a big bureaucratic agency that can abuse its authority and firms can engage in rent seeking and so forth. So if you think that regulation is too difficult, you might prefer to just split firms up using antitrust judgments or make it illegal for them to grow or to merge or whatever. So there's a lot of sort of the key pieces of legislation in the US are the Sherman Act from 1890, which prohibits what they called restraints on trade. The Clayton Act, which outlaws charging different prices to different customers or making a customer buy one thing if they want as a package, if they want to buy something else, the Robinson-Patman Act, which outlaws so-called predatory pricing. Other countries, the EU has competition policy. Of course, it's newer. These laws come from the 60s when the European Commission was becoming established but they essentially imposed the same kinds of restrictions on firms as antitrust laws in the US. Same thing in many other countries. Well, you can imagine there are a whole host of problems with using antitrust action to deal with so-called monopolies. So one problem is, well, what's the relevant market? So again, people say, well, Google has a monopoly on search. Common statement. Okay, well, I mean, as we've already pointed out, I mean, it doesn't technically have a monopoly on searching on the internet because there are other search engines.  There's a lot of search engines. Now, are they as good as Google? Well, I mean, I don't know. That's not something a court can easily determine, right? But clearly there exist other search engines. In fact, I mean, there are other ways to get information too. Oh, I want to eat pizza at some place downtown. What time do they close? Google, ch-ch-ch-ch-ch. Well, guess what? For the younger people in the room, there was a day when we didn't have these things. And if you wanted to find out when the restaurant closed, you would like call on a traditional phone and say, hey, what time do you close?  Well, we had these big, thick books called telephone directories, where you just go look up the name of the pizza parlor, look up the number, call. Okay, yeah, that's not, I prefer doing it this way. I prefer Googling it, but that doesn't mean there aren't other ways to get information besides Google. So what counts? There's a famous antitrust case in the U.S., the 1960s called the Brown shoe case. That's an old Brown shoe advertisement where the court said, well, basically the relevant market for determining monopolistic practices is whatever affects people's behavior in any market. In other words, the defense Brown shoe is arguing that they sell their products on a national market, right? And so they were competing with all the other firms that sell in national markets. But the Justice Department said, no, no, the relevant market is each individual town. So if you have 50% of the market share in one particular town, then you have monopoly power. Even if you only have 5% of the national market. So I mean, is this sort of an arbitrary decision on the part of the court as to what is a relevant geographic market? There's a lot of issues that the sort of approach is way too static, doesn't take into account what the future of the market might be.  In fact, private parties in the US can actually file antitrust complaints against their rivals, their competitors. There's obvious potential for abuse in that case. Antitrust cases take a long time to resolve. There's a famous example in the 1960s and 70s, the US claimed that IBM was a monopolist and there's this lengthy antitrust trial that went on for more than a decade. And by the end of the trial, the market that IBM was alleged to have monopolized, namely the mainframe computer market, it pretty much disappeared. So they just dropped the case because there was no longer anything to talk about. But from a kind of a philosophical point of view, big problem with antitrust law is that it's all kind of what the legal, the jurists call ex post facto rules. Meaning there's no way to know ex ante if a given behavior is or is not a violation of the antitrust law. You only find that ex post based on whether you win or lose the antitrust case.  Well, yeah, maybe I'll be able to get away with it. At least in principle, I know punching somebody in the face is not legal. But I sell pizzas for 10 bucks a pizza, I wanna charge 11 bucks a pizza, is that legal? There's no way to know. I mean, that could be an antitrust violation. There's no way to find out until you try it and get sued by the Justice Department. There's a great book that you gotta look at by, I think we have it downstairs. It's a comic book, sorry, a graphic novel called Tom Smith and His Incredible Bread Machine. It's kind of a funny book on about a fictional entrepreneur who invents this magical bread machine then the government tries to shut him down. There's a great scene where the antitrust officials are berating him with the following text. You're gouging on your prices if you charge more than the rest, right? So price gouging is when you charge a price that's too high. But it's unfair competition if you think you can charge less. That'd be predatory pricing, right? A second point we would like to make to help avoid confusion, don't try to charge the same amount, that would be collusion. Okay, so no matter what price you charge, you could conceivably be violating the antitrust laws. Okay, so to summarize from an Austrian point of view, competition should be understood as a process of rivalry among entrepreneurs who are free to compete as they see fit within some kind of illegal framework. And that might result in markets with just a few firms or markets with a lot of firms and they might be big or small. Firms might earn high profits, they might make losses. There's all kinds of things that could be going on in those markets. As long as the government is not granting special favor to some entrepreneurs over others, we can say that that market is competitive. And it follows that attempts by the state to sort of limit monopoly power or monopoly privilege cannot improve wellbeing, cannot improve market performance and so forth. The best government policy towards monopoly is don't give any out, right? Don't create monopolies and then we don't have a monopoly problem. Thanks.


Hey there friends, thanks for checking in welcome to another video in my series called one handgun one shotgun and one rifle The hang-up that I'm featuring today is a sig sour p3 20x Compact when I feature this gun. There are so many people that say I have that gun I love it. I carry it. It is so reliable and I have to say since I have owned it I have really enjoyed it 15 round magazine 3.6 inch barrel front serrations a rail a Flat-face trigger that breaks right around five and a quarter pounds and a reset right there X-ray three front dot night sight regular night sights in the back. I think they're siglight night sights but an amazing striker fire Nine millimeter it's a it's a step above the average one or nine and my Estimation let's take a few shots here Yeah, it's a sweet shooter The X carry is actually larger the X compact It's actually smaller. I think sig may have reversed that in the opposite direction Nonetheless, this was a hot gun last year so hot and I think it remains so today Now let's get to that shotgun the shotgun that I'm featuring today is a Stoker M 3500 semi-auto 12 gauge with a 26 inch vet rib barrel a Fiber optic front sight a finish called real tree looks beautiful a honeycomb recoil pad Cross bolt safety the top of the receiver is drilled and tapped for a rail if you throw an optic on there its current capacity is three because The owner of this was hunting with it and it's just a very smooth shooting shotgun and He purchases for under $700 and he loves it and said that it never failed on him So Stoker makes some great shotguns and this M 3500 is Not an exception. Let's go ahead and the bolt locks back Very nice shotgun. It's good-looking smooth performing and Certainly gets the job done the choke here is a timber ridge choke It's a full choke and that's a new and up-and-coming company that we're probably gonna hear More about in the future now. Let's get to that rifle The rifle that I have is actually a 9 millimeter pistol But CMMG makes a resolute model that's the exact same with a 16-inch barrel So we're gonna pretend that this is that or that is this but what we have here is a CMMG Banshee MK 17 with a beautiful Titanium Cerakote finish check that out what makes this unique is that it takes six hour p3 20 mags Which is why I brought out the p3 20 X compact as the handgun because it will accept the same mags And I really like that p3 20 line is huge. It's popular and it's growing 21 round mags, but check that out 8 inch barrel CMMG SV break It does have the rip brace it allows you to rip it out quickly Keep it in shortest form to rip it out and you're good to go magpul grip aluminum M lock hangar 1913 rail cross single-stage CMMG trigger and a beautiful looking and functioning Pistol that I love the optic here is a vortex spitfire Let's go ahead and take a few shots. Let's see how this does Wow This never gets old huge fan of CMMG and I love this Banshee MK 17 and if you love this and you want a carbine Check out the resolute line. You can get this exact same thing with a 16 inch barrel and it is just as awesome Thanks for watching another video in my series called one handgun one shotgun and one rifle If you like videos like this, please subscribe and share. I always appreciate thumbs up button Thanks for watching and you guys be safe


Give us a brief rundown of what conspiracy means to you and what role it plays in your work. Well, the role it plays in my work, again, is what I see primarily as a function of sympathetic magic. Now, this process is the process in which symbolism and means are and icons are manipulated intentionally in order to achieve them as a result. And I think, you know, a phrase that Gordon and I use a lot, Gordon White of Rune's Stoop, use a lot is getting their attention. So in other words, I think where a lot of people tend to look at this symbolism that they see in media as purely a function of conspiracy, that it's all, you know, that we're all being sort of entrained in a sort of MK Ultra, Naomi, you know, Chatter, Bluebird, whatever program you choose, that it's all a function of some sort of program. And I think in there is a tremendous amount of entrainment going on. And that's something that I've been trying very hard to detail, because I think that we're seeing what I call emerging archetypal dominance. And that is that new symbols are being fed. People are still looking for like these sort of 15th century Rosa Crucian, Masonic, Templar, I mean, all those sort of symbolism being fed into the culture. And I think what that material was and what its function was and why we saw so much it in rap and music video and in high fashion is that that was part of the entrainment process. And it's basically getting people to respond to symbolism in a way that they hadn't consciously. So it's making what had been subconscious and unconscious conscious, because I think that there is an effort to change mass consciousness in a way that I don't believe this world has ever seen. And I think that, you know, I don't think this is speculation or conspiracy theory per se. I mean, you know, there are papers published on this through academic journals and psychology and science journals. And, you know, we're really seeing because of this unprecedented extent of data that we're forced to process every minute of our lives, that consciousness is changing.


Okay, so today I'm going to be talking about distribution or equidistribution of nil sequences. I'll remind you just quickly what a nil sequence is shortly. But really I'm going to be talking about generalizations of a very, very well-known fact. So generalizations and quantitative variants of the following very well-known fact, which is that if theta is a real number, then the sequence of values theta, 2 theta, 3 theta, etc., is equidistributed mod 1 if and only if theta is irrational. So I'm going to be generalizing that to polynomial sequences on arbitrary quotients g mod gamma. Here I've just got r modulo z. And I'm also going to be interested not in the whole sequence up to infinity, but in finite segments of it. And all of that with a mind to applications next time and the time after.  And recall, we have a simply connected nil-potent Lie group g, which has on it a proper filtration g-bullet. And that's a collection of, it's a collection of closed Lie subgroups of g, which satisfy this commutator inclusion property that I've mentioned before. And then I define this notion of a polynomial sequence that we defined in several different ways, which I didn't show to be equivalent, but I stated it. The notion of polynomial sequence p of n, p from z to g. So I won't go over that again. I did it last time. And you don't really need to know what the general definition is this time. So the basic question I'm going to be interested in is, when is, let me take a lattice gamma, let gamma be a lattice. If I have a finite set of values of that polynomial, so when is p of n from n equals one to big n close to equal distributed in the quotient gamma mod g? And what does this even mean? So that's the question I'm going to be interested in. Well to motivate this, maybe I'll give you a few examples. I've already had one. So let me first of all say that the example I put up there actually generalizes, this is perhaps a less well-known fact, it generalizes to arbitrary polynomial sequences. So g equals r and gamma equals z. And if p is just any polynomial. So I think I briefly mentioned last time how to make the traditional notion of a polynomial compatible with this slightly fancy notion to do with filtrations. If the polynomial has degree d, you just take the first d elements of the filtration to equal the whole group, which is r in this case. So if p is any polynomial, then p of n is the whole sequence, is equidistributed. If and only if the polynomial has at least one irrational coefficient, other than the constant term, other than the constant term. So this is not so trivial to prove. I may say a few words about it later. So the proof uses vials inequality, which I will mention later. So vials inequality will tell you that if the polynomial is not equidistributed, then the lead coefficient is irrational. And then you'd have to iterate down to show that all of the other coefficients are irrational as well. So uses vials inequality and an iteration. And there are theorems in a Goddick theory. So a theorem of Leon Green, no relation of minus should point out. And this is about equidistribution of flows on nilpotent groups. So if g is a nilpotent, so a simply connected nilpotent lead group. And if p is a linear sequence, so p of n is a to the n for some a. So that qualifies as a polynomial sequence with the definition I've given. Then the conclusion is, so p of n, n equals 1 to infinity, is equidistributed. What does equidistributed mean? Well, I'll be going over that shortly. But let's just say for the moment, with respect to the Haar measure, so with respect to Haar measure on gamma mod g. Well, if and only if the only way that a sequence can fail to be equidistributed is if it fails to be equidistributed in an abelian projection of g. So if and only if the abelian projection of p fails to be equidistributed. So what I mean by that is that you have g and you have a natural projection to g mod its commutator. And the sequence p projects there. So this will be isomorphic to just a Euclidean torus. And when you project p, it will be isomorphic to a certain rotation. So the p projects to a map. Pi composes with p. So this induces a map on a torus. So let me give an example with the Heisenberg group. So if and only it is equidistributed, if and only if the abelian projection is equidistributed, exactly. Thank you. I think it's best if I give an example. So example, if g is the Heisenberg with the usual lattice, and if a is just some element, so if a is 1, 1, 1 alpha beta gamma, then the flow that's induced on the two-dimensional torus by that is just rotation by alpha and beta. So the induced map on the torus, r mod z squared, is given by p tilde of n is alpha n beta n. So you have Leon Green's theorem is the non-trivial fact that the sequence a to the n in the nilpatent group is equidistributed if and only if this sequence is equidistributed in the torus, which by Kroniker's theorem is the same as saying that alpha and beta are independent over the rationals. So alpha, beta and 1 are independent over the rationals. By Kroniker's theorem, this is equidistributed if and only if 1 alpha and beta are linearly independent over q. So that's a complete criterion for when a nil rotation on the Heisenberg group is equidistributed. So I'm really going to be interested in quantitative variants of this, and even to say what I mean, I should give a bit more discussion. So are there any questions so far? So quantitative equidistribution. If you want to make things quantitative, you have to quantify the notion of close. So let me introduce a parameter delta greater than or equal to zero be a parameter. Well, if somebody asked you how should I define equidistributed on the circle, probably the first thing I would say is it's a sequence that spends the right portion of time in each interval. It's slightly problematic to use that as a definition, because what should be the analog of interval in a more general group? So we don't think of it quite like that, but we think of the interval, the characteristic function of the interval as a function, as it is, and then equal distribution is saying that averages of our sequence along a function are close to the integral of that function. So we say that a sequence p of n, so it will always be for us a polynomial sequence taking values, well, the sequence p of n, is delta equidistributed in gamma mod g if, so when you look at the average of this sequence weighted by any function on, so this phi will be a function on g mod gamma, and then you compare that to the integral with respect to harm measure. Then that's at most delta times an appropriate norm of the function. I'll tell you which norm in a moment we're here for every, let's say for every smooth, automorphic function on g. So here we put the harm measure on gamma mod g induced from the unique harm measure on g, and it normalized in such a way that the total volume of this quotient is one. So the measure of the whole space is one. And these harm measures are not too difficult to understand in the Heisenberg group, it's actually just dx dy dz. So in Heisenberg the integral is simply with respect to dx dy dz. So the harm measure is just the product of the harm measures in the three coordinates. So you won't lose much by thinking about that case or even just the circle case for now. It doesn't matter hugely which norm you take, I mean if you change the norm it will change the notion that you're dealing with, but in relatively minor ways. So for technical convenience it's actually useful to have our norm to be one of those Sobolev norms that I mentioned last time. So we'll take the norm. So you just take it to be a sufficiently high Sobolev norm, and I think I found that the Sobolev norm with 2 to the s times the dimension of g derivatives was enough. But that's really a technical matter. In fact you could just look at first derivatives of phi and every function with bounded first derivatives is basically Lipschitz, or it is Lipschitz, and it can easily be approximated by much smoother functions. So that's really just a technical point. Anyway the basic idea here is simply that you're comparing space averages and time averages and that's how you decide whether a sequence is close to equidistributed. Questions on this? So a crucial result on, well I suppose the main result that I want to talk about is a more or less complete statement about when a polynomial sequence on g mod gamma is equidistributed in this sense. And it generalizes really the examples that I showed you above. So the main result, a generalization of the above examples to, well first of all, two polynomial sequences on gamma mod g and with this quantitative notion of distribution. Now maybe I should say, well I should say, so Leon Green's theorem is valid for any simply connected nilperton group and for any linear sequence on that group. If you instead want a polynomial sequence, that's a more general result. But it turns out that actually a polynomial sequence can be lifted to a linear sequence on a much bigger group. So actually you get the polynomial equal distribution result for free from the linear one. But for various reasons that's not, that argument fails you in this quantitative setting. So I thought I'd mention it but I don't want to place any emphasis on it. So what I do want to do is state precisely this main result and then I'll sketch some ideas of the proof. So here is a reasonably precise statement and it seems to be a bit of a theme in these lectures that the most I can hope to do with precision is state results. I guess that's, sometimes things are difficult. So theorem, let the notation be as above. Let's suppose that p fails to be delta equidistributed from 1 up to n. Well then there's an abelian explanation for that and I'll write down what it is. There is a horizontal character. So I'll introduce a few terms that I will explain later. A horizontal character eta from g to the reals, mod z, such that what we call the smoothness norm of the abelianization of p is bounded. So this bounded by delta to the minus a constant where the constant depends on the group. So I have a lot of things here to explain. But if you don't wish to take too much notice of my explanations, this is an abelian obstruction to a sequence being equidistributed. So here pi is projection from g to g mod, actually, sorry, with the way I've defined it here I don't need the pi. This will automatically be a, so I just have to tell you what a horizontal character is. So here horizontal character just means it's a homomorphism. So it will automatically annihilate the commutator subgroup g brackets g and it also is supposed to annihilate gamma and the smoothness norm. So thus eta composed with p of n, well it's going to be an r mod z valued polynomial. So it's a polynomial map. Let's call it f from the integers to r mod z. And so I can just write it in the usual way that I write polynomials. So this will have the form f of n is alpha naught plus alpha 1n plus up to alpha d n to the d.  So to say that, so the statement and that f in smoothness norm c infinity of n is at most m means, well it means that those coefficients are very close to integers. It means that alpha j in r mod z, which is the distance from alpha j to an integer, is at most m over n to the j for every j. So that's a lot to take in. Let me tell you at least how it specializes down to just even some abelian settings.  Actually it's telling you that they're close to being integers. To somehow, that's enough because you can always multiply eta through by an appropriate integer to make things that are close to rationals close to integers. Let me give an example. An example just with g equals r and gamma equals z. The sequence theta n squared from n equals one to n fails to be delta equal distributed if and only if theta is very, very close to a rational. So if and only if there is a q in the integers such that the distance from q theta to the nearest integer is bounded by delta to the minus constant over n squared. Where is the q coming from? Well multiplication by an integer q is the same thing as one of these horizontal characters. Let me just write that. So eta is multiplication by q. So this is a known result. This follows from vials inequality, but this is not the usual way of phrasing vials inequality. Actually it's the way that I prefer to, when I'm lecturing on vials inequality, I prefer to state it this way. I find the usual way a little bit confusing. I don't know if anyone's familiar with the statement of the inequality. It's a statement about certain exponential sums with polynomial phases, but it's really saying precisely this. So a polynomial sequence and when it's equal distributed. So does anyone notice something that's lacking in that statement? Well, sorry? It's bound on q. As stated, it has no content whatsoever, right? Because every theta by just by Dirichlet's pigeonhole principle, for any theta you can find a q, it might have to be very big, such that theta times q is close to an integer. So I additionally need a quantification to give this content. We also need to place a bound on the complexity of this horizontal character eta, that a certain complexity quantity, so the complexity of eta. And as before, this has got to be measured with respect to a basis on the Lie algebra, curly b, is bounded. So I don't want to go over again what the basis was and how you define exactly this notion of complexity. But in this particular example, it just degenerates to the size of q. So in the example just given, it asserts a bound on q, mod q. Also q should not be zero, otherwise it's again completely trivial. So it asserts a bound on the size of q, in this case, well, the size of q is again bounded by delta to some power. So to state the theorem precisely, I would need to include also some bound on the complexity of this horizontal character. So let me just write explicitly what the contents of this theorem is in one phrase. So p of n can only fail to be equidistributed for a billion reasons. Probably all of a sudden the power of delta on the left hand side delta to the k equals 100. Power of delta where? On the left hand side of the equivalence because you can replace delta by delta. Ah, yeah, yeah. Let me just, okay, that's a good point. The notion of equivalence in this sort of discrete setting is always a bit, you don't quite get back to where you started from. So there is an if and only if, but if you go that way and then back, delta will have become delta to some bigger power. So let me just phrase it like that for now. Here's the basic idea. A polynomial sequence, the only way it fails to be equidistributed is for a billion reasons. Now I want to say a little bit about the proof, obviously not very much because it's a, this is a really difficult theorem to prove, but the proof uses a trick, well it's more than a trick, a technique that is the same technique that's used in the proof of vials inequality and that's called the van der corporate lemma. So the main idea of the proof. So following vile, and this is not, so this follows vile which is a paper in analytic number theory. It's not the same thing that Leon Green does in his paper, but by following vile we will, we use what's called the van der corporate lemma. So the van der corporate lemma asserts a relation between a sequence being equidistributed, well the original van der corporate lemma, so the original version of which, asserted a link between the distribution of a sequence, the uniform distribution of a sequence un mod 1 and its derivatives, which are un plus h minus un. Specifically if all of the derivatives are equidistributed then so is the sequence. So that would actually immediately give you qualitative equidistribution results about theta n squared, because the derivatives will then be linear sequences and then you can just apply the known results on linear sequences. Here is our take on van der corporate inequality, which is very standard. Suppose you have just any bounded sequence, so suppose that a n, n equals 1 to n, are complex numbers of norm at most 1 and suppose that their average is not close to 0. So suppose the expected value of a of n is at least delta, then the conclusion is that many of the derivatives have the same property. So I'm going to state a precise version even though I'm certainly not going to prove a precise version. So suppose that big h, slightly technical condition, big h lies between, big h is neither incredibly tiny nor quite as big as n, then we have that the average value of many of the derivatives is at least delta squared over 4 for many values of h. Remember how many? The 4 and the 32 clearly don't matter, so for at least delta squared over 32 times h values of h. Sometimes I find it less confusing, I don't know if you agree to put in explicit constants rather than unspecified absolute constants when they're not too big. So the statement is if you've got some bias in a sequence then also many of its derivatives are biased. Now let me sketch the proof. Well the idea is that if h is small and that's going to be guaranteed by this inequality then averaging the average of a n is very close to the average of a n shifted by h. I mean they're averaging essentially the same thing except you've got some slight edge effects. So many of those averages are well, so then it's basically the Cauchy-Schwarz inequality. So you can choose just some phases, so let's choose some unit complex numbers, zh expected value n less than or equal to n of a n plus h is bigger than delta over 2 let's say for an appropriate phase zh just to make it real. So if you average that over h, so the average over h less than h and the average over n less than or equal to n of z of h a of n plus h is also at least delta over 2 and now you apply Cauchy-Schwarz. So by Cauchy the average over n of the average over h a n plus h squared is at least delta squared over 4 and if you expand that out so that implies that the average over h1 and h2 less than or equal to h of the average over n less than or equal to n of a n plus h1 a n plus h2 bar is at least delta squared over 4. Well you can see that with a bit of messing around you'll get the statement that I wanted. Every average on the inside here is basically like the average in the statement of van der Korpen's inequality so this is roughly the expected value over n less than or equal to n of a n a n plus h1 minus h2 bar and then you just have to count how many times each h1 minus h2 occurs so etc. But that's the basic idea is just one application of Cauchy-Schwarz after you've done a shift. Questions? The other key idea is also a key idea a very famous idea that is due to Vile which is that to understand equal distribution you could you should expand into Fourier series. So the other crucial idea is Vile's idea of proving equal distribution by expanding into Fourier series or put another way we had a definition of equal distribution which sadly I've erased but the definition was about an arbitrary function phi and you just need to check that definition on a basis of well l2 of gamma mod g so check the definition so check the definition of equal distribution only for phi lying in a basis of l2. Now of course it's not quite enough to just check things if you're dealing with a quantitative questions you can't just check things for a basis and then hope that the same statement will hold when you add basis elements together but as long as your way of decomposing into these basis elements is suitably effective which it will be if the functions are highly smooth then that's more or less the case. So what Vile did famously in the case of r mod z it's enough to check it so it's enough to handle phi of t being e to the 2 pi i mt for m in the integers and as I said you do lose a little bit in the delta so this causes some slight degradation some slight change in delta but just a power. So I'll go through that in the case of null sequences in just a second but let me explain how it works already in the case of say theta n squared. So if you want to test the distribution of theta n squared on the circle it's enough to test it for e to the 2 pi i theta n squared and if that average happened to be large then you could apply the van der koper inequality and get that many derivatives of that have to have large average but a derivative of e to the 2 pi i theta n squared is just a linear phase function which is just a geometric series so it's quite easy to evaluate that. So what's the appropriate variant of that of this Fourier expansion in gamma mod g g mod gamma where g is a nilpotent group well of course that's a that's a very deep question in general but actually it turns out we can get away with a fairly simple answer to that we're not going to look at a sophisticated way of decomposing this space it's actually just enough to do a Fourier expansion in the vertical components so in the general case a similar trick works to test equal distribution for an arbitrary phi you decompose it into automorphic functions with a vertical frequency so to test equal distribution against an arbitrary phi in c infinity of gamma mod g it's enough to handle the case when phi has a vertical frequency psi which let me remind you means that phi of if you twist x by an element of the last group in the filtration then it has a natural transformation property under psi and the reason for that simply is that there is um you can do a Fourier expansion in the direction of this group g so as I mentioned last time uh phi can be expanded as a sum of automorphic functions with a vertical frequency and of course all of this would have to be done quantitatively but I shan't bore you with the details of that so suppose you've done that and suppose that you have now a function that additionally has a vertical frequency so suppose that psi is a non-trivial frequency then that that means just as with the non-trivial exponentials on the circle the integral of this function with respect to the harm measure because of the vertical oscillation it just cancels out and you get zero so what does equal what does failure of equal distribution mean in this case so if p of n from n equals one to n fails to be delta equal distributed with respect to uh this particular automorphic function with a vertical frequency then it just means that the average is at least delta well actually at least delta times the smoothness norm of this function but I'll suppress that so this is something that we can apply the van der corporate lemma to and by van der corporate there are many h such that the same average but now with an additional term of phi of p of n plus h is at least some constant times delta squared now I don't know if anybody remembers what happened last week but you can see that something very similar to what happened in the proof of vials inequality has happened here so for vials inequality van der corporate took a quadratic phase function and by differentiating it made it linear and what is the derivative of a nil sequence while I spent a whole hour explaining that that is a nil sequence of class one less so we can turn this into an induction by this device of taking derivatives so by the main discussion by the discussion of um I think it was mainly in lecture three uh the object here phi psi p of n phi psi p of n plus h bar for fixed h so is for fixed h a nil sequence of class at most s minus one so what this means is that you can imply you can hope to prove the main theorem by induction on the class so this offers the possibility of proceeding by induction on s so this is very special to the case of these nil flows one can consider flows on g mod gamma in other cases for example where g is sl2r um gammas a subgroup but the same sort of trick just wouldn't work I mean you'd get this would not be a simpler object in fact it would be a worse object sitting on g cross g but in the nil potent case we do have this simplification so that's the main idea of the proof but you still have to actually carry out the induction so carrying out this induction is uh rather difficult and I just want to show you just give you a glimpse of what happens in the heisenberg case so you can see the way in which it's difficult um so let me actually revert completely to the notation of last time so let's set chi of n to be this nil sequence and then in the notation I had last time delta sub h of chi is precisely the derivative so last time I said that this derivative is a nil sequence of class s minus one well actually let me let me specialize also to the case of the heisenberg just for illustration so g is the heisenberg and um and the polynomial is just a to the n for some n linear so I said last time last time we saw that the derivative delta h chi of n can be thought of as a nil sequence of class one and I wrote down precisely what those actually I wrote down the general form so here p sub h delta of n is p of n plus h so it's p of n p of n plus h times p of naught p of h inverse and that's actually in the linear case it's just the same it's just the diagonal flow a to the n a to the n in the linear case if you calculate very easy calculation and the automorphic function phi phi sub h box of n was phi times phi bar of x y p naught p h which in my case is phi of x times phi of y times a to the h and here we come across a serious issue which I alluded to last time so I'm trying to prove a statement about equal distribution by induction but the statement I'm trying to prove is quantitative it cares about how smooth my functions phi are and how lipschitz they are but unfortunately this a to the h here could be huge I said nothing about how a is bounded this could be an absolutely enormous element in the group g so there's no guarantee that this automorphic function has any nice control at all so a could be huge a to the h could be huge so no control on phi h square so the induction won't work unless you do something to get around that issue and what you need to do is to insert a conjugation by elements of the lattice gamma so to get around this we can conjugate by elements of gamma cross gamma so here and maybe I'll just show you how to modify this to do that so you can put a you can conjugate this by an arbitrary gamma gamma h and this has the effect of adjusting the automorphic function in here so put some maybe to explain this I did a computation last time explaining why the derivative of a null sequence could be written in this form it turned out that there's extra flexibility inside there that I didn't make use of you can do this conjugation in any way you like and you have a different representation of the derivative in this form and so this will then become delta h I can take gamma naught equals zero gamma naught is the identity so I get to multiply a to the h by any element of the lattice that I like on the left and by doing that I can make it small so by choosing choose gamma sub h so that gamma sub h a to the h lies in the fundamental domain so in particular we'll have all of its matrix entries bounded by one and so when you do that you do then recover control on this automorphic function here unfortunately you've made the problem more complicated so you had to do this to make get yourself a derivative function that's got bounded smoothness norms but you've made the polynomial the derivative of the polynomial more complicated so p sub h is now while it's rotation by a by the diagonal a a conjugated so it's delta sub h gamma sub h a gamma sub h inverse to the n so now you can imply the inductive hypothesis this is just a rotation on a torus it's a rotation on a torus so I introduced this torus last time it's what you get by applying the constructions I showed you last time on the Heisenberg and it's a three-dimensional Euclidean torus and so by Kroniker's theorem we have essentially the coordinates of this vector a and the conjugate of a fail to be independent over q so the coordinates of a gamma h a gamma h inverse well reduced to lie in this torus so reduced mod g2 diagonal fail to be independent over q for many values of h so it's an abelian problem but it's a rather complicated one so you have to somehow go from that assertion to saying something about the coordinates of a itself it's a difficult matter to go from this to a statement about a but it can be done um I think it's well I'm not going to go into any more details the the the paper in which this is done is difficult but at least you're reassured that you kind of know it has to work if the theorem that we're aiming for about distribution of nil sequences is correct and that's because this van der korp is almost an if and only if so it's not you haven't really lost anything apart from a few powers of delta in going through all of this machinery the expansion into Fourier series again it's more or less an if and only if so this is a it's kind of a weird sort of integration problem you've taken derivatives and you have to deduce it was easy to take derivatives but the statement you ended up with when you did that is quite difficult to integrate back again and I should say that it's the same problem with the proof of the inverse conjectures for the Gauss norms I showed you last time why why nil sequences are obstructions to Gauss uniformity by induction so I said if a function correlates with a nil sequence chi many of its derivatives correlate with derivatives and then it I was done by induction but to get the converse statement is much harder so many of the derivatives of f correlate with a nil sequence but you have to then deduce from that that f itself correlates with a nil sequence so it's a similar sort of issue that's difficult there this is not to use it it is all over the place actually yes for in the case of the u4 norm and higher this is used everywhere I mean I didn't use the inverse conjecture at any point here this is purely an almost an algebraic fact really I've said nothing about Gauss norms for the u4 norm and higher it's needed this is really this is somehow key to the entire theory of higher order Fourier analysis and nil sequences it's also needed crucially in showing that well tomorrow I'll be talking a bit about the primes and there you need to show that the Möbius function is orthogonal to these nil sequences and again this theorem is a crucial ingredient there so it's somehow quite fundamental um well just to finish I want to say two things about how this is applied so why I haven't really said anything about why I should care about equal distribution of nil sequences so why do I care about this particular notion equal distribution of polynomial sequences on gamma mod g so there are two reasons for that and the first one is that essentially every sequence is equidistributed so every p of n is almost equidistributed I'll say a little more about what that means in a moment um but it means that p an arbitrary polynomial sequence can be decomposed into some very benign pieces a periodic piece and a very smooth piece plus an equidistributed piece so somehow the equidistributed case is pretty general and then secondly if p of n is equidistributed then we can we can count various linear configurations so if I have time I'll give you an example so eg the average of I can count say four term progressions involving nil sequences let me make this r p of n plus two r phi of p of n plus three r and it will be I get a very algebraic answer it's an integral of phi to the four over a certain subgroup called the whole potresco group h p four of g modulo so this where h p four of g which is a subgroup of g cross g cross g cross g is a certain group explicit group called the whole potresco group the whole potresco four group um and actually that this generalizes there's nothing special about arithmetic progressions there's a corresponding result for any linear forms which is called the leadman group um let me tell you what the whole potresco group is because we've already seen it in action eg when g is so when g is r but with the filtration so I'm going to put a degree across two filtration on g then the whole potresco four group h p four of g is it's a subgroup of r four and it's the set of all topples x naught x one x two x three such that x naught minus three x one plus three x two plus x three is equal to zero and we've seen that before um because I've already we know an example of a polynomial sequence here would just be theta n squared um but the example I showed you before was where theta was root two so let's give that example and we noticed before that there's a constraint amongst four term progressions uh in this function so I mentioned that when I was showing you that traditional Fourier analysis is not enough to handle four term progressions and what this theorem is saying is that essentially that's the only constraint so it's saying that the four term progressions or inside this function equal distribute over the set of all four topples satisfying this constraint so um in a sense so n squared root two minus n three n plus d squared root two plus three n plus two d squared root two minus n plus three d squared root two equal zero is the only constraint on four term progressions for ap's in p of n so these equal distributed sequences are very nice actually I lied to you slightly uh that's false you actually need a slightly more elaborate notion a stronger notion of equal distributed called irrational it's quite technical and I I don't feel inclined to say what it is here in many cases for example for for these linear sequences on the Heisenberg group the two notions coincide so to finish today I'm going to explain what I mean by this statement what do I mean by every polynomial sequence is almost equal distributed uh well this is basically a generalization of what's called the Hardy-Littlewood method so Hardy and Littlewood were concerned with well things like polynomial phases on the circle when they were thinking about wearing's problem and they decomposed those phases or the coefficients of those phases into major and minor arcs and um so in other words they basically said every every numbers either essentially rational or it's not rational it's irrational and that's essentially what's generalized by this assertion here so let me explain point one is uh really a generalization of the Hardy-Littlewood idea of decomposing into major and minor arcs and in fact the heart this Hardy-Littlewood idea can be seen as the special case where g is the circle a g is r of this so a more precise statement so given an arbitrary polynomial sequence I can write it as a product so I can write p as epsilon times p prime times gamma prime this is a point wise product of polynomial sequences uh where epsilon epsilon just varies very slowly smooth so that means something like uh the I mean some on on the group measured on the group in some way it's just doesn't vary very much so this means essentially that you can always use some standard tricks in analytic number theory like partial summation to just ignore it and then gamma is rational and that means that um basically that gamma is periodic modulate the lattice for some small q and again in practice that means you can get rid of gamma just by or just by splitting into sub progressions of common difference q and then finally p dashed is equal distributed is highly equal distributed somewhere um not necessarily in g I mean the p could just be trivial p could be identically equal to it just could be the identity always and then obviously you're going to have a hard time saying that anything related to it is equal distributed on g but p primed of n is highly equal distributed somewhere on some g primed mod gamma with g primed contained in g a closed connected subgroup and as with all of these statements a proper quantification of this is quite technical um you need to say what you want by highly equidistributed and you get to choose it to an extent at the at a certain cost um and then you get to control the complexity of the g primed etc etc so there is a much more precise statement that needs to be made but this is the this is what's true in spirit so in the example I mentioned where p is constant this decomposition is trivial so epsilon is also constant gamma is constant and p primed is constant and it's highly equidistributed in where g primed is identically one the identity um and then the other extreme would be where p is already equidistributed and then you could take epsilon and gamma to be trivial so just to finish let me explain how is this proven and this point is why this is the reason that we have to deal with polynomial sequences rather than merely linear ones so the way this is proven is by induction on downwards induction on the dimension so if p is already equidistributed then you've got nothing to do stop otherwise the main theorem I've been talking about today tells you that there's an abelian explanation for that fact and what this means is that p is basically near it sits inside a proper subgroup so you then get to induct downwards on dimension and that will eventually stop so the proof is by downwards induction on dimension using the equidistribution theorem and let me give you an example of what can happen when you do that so e g again with the Heisenberg suppose that you take p of n to be a to the n a linear sequence where a is one one one theta one zero and here theta is of size about n to the minus three halves so you can compute what a to the n is a to the n is one theta n one one n and then a half theta times n times n minus one now a n a to the n n equals one to n is not close to equidistributed and we know by the theorem that there must be an abelian explanation for that but in this case it's staring us in the face because the x-coordinate never really leaves zero so it's visibly not it not close to equidistributed because theta of n is stuck near very near to zero and so if I follow through this idea of in downwards induction on the dimension let's restrict the whole setting to the group where x is zero but then a to the n is no longer a linear sequence it's then a polynomial sequence with those entries so this can turn a linear sequence so p of n turns into a polynomial sequence upon restriction to x equals zero so in fact it's a polynomial sequence on an abelian torus in that case so what this point shows is that you have to work in this category of polynomial sequences in order to be able to make any of this work just the linear sequences don't have enough closure they don't have this it's essentially the group property of polynomial sequences again I mean a to the n times b to the n is not of the form c to the n in the not abelian world okay so that's what I'm going to say about that there are two lectures left and they will only tangentially rely on this and not on the details of this I think next time I'll talk about semi-radius theorem cases of that and some variants of that and then last lecture I will talk about applications to the primes but that's all for today thanks um questions at all yeah yes and you to do that you need a two variable version of the equal distribution theorem so instead of just a sequence index by n you need a sequence index by n and d and actually so the paper in which these results were proven is this paper quantitative behavior of polynomial orbits on no manifolds annals of mass 2000 and well it says 2012 but this paper was written in 2006 and recently we discovered that this was wrong the proof of that multi parameter version so it's now being corrected but it was it was wrong for six years actually um hence actually all of the in a sense the proof of the inverse conjectures was I mean it was wrong but fixable but yeah that's what's needed here


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