---
abstract: |
    We show that if $f : {\mathbb{A}}_{\bar{{\mathbb{Q}}}}^r \to {\mathbb{A}}_{\bar{{\mathbb{Q}}}}^r$ is a regular self-map and $P \in {\mathbb{A}}^r(\bar{{\mathbb{Q}}})$ has $\limsup_{n \in {\mathbb{N}}} \frac{\log{h_{\mathrm{aff}}(f^nP)}}{\log{n}}
    < 1/r$, where $h_{\textrm{aff}}$ is the affine logarithmic Weil height, then ${\mathbb{N}}$ partitions into a finite set and finitely many full arithmetic progressions, on each of which the coordinates of $f^nP$ are polynomials in $n$.

    In particular, if $(f^nP)_{n \in {\mathbb{N}}}$ is a Zariski-dense orbit, then either $r = 1$ and $f$ is of the shape $t \mapsto \zeta t + c$, $\zeta \in \mu_{\infty}$, or else $\limsup_{n \in {\mathbb{N}}} \frac{\log{h_{\mathrm{aff}}(f^nP)}}{\log{n}} \geq 1/r$. This inequality is the exponential improvement of the trivial lower bound obtained from counting the points of bounded height in ${\mathbb{A}}^r(K)$.
---

[A lower bound on the orbit growth\
of a regular self-map of affine space]{}

Vesselin Dimitrov\

`vesselin.dimitrov@yale.edu`

[**1. Introduction**]{}

[**1.1.** ]{} In the appendix to our preprint [@prelim] we formulated a precise conjectural criterion for the algebraicity of a formal function on a projective curve over a global field. For the case of the projective line, this criterion generalizes simultaneously the classical Pólya-Bertrandias criterion (cf. Amice [@amice], Ch. 5), on the one hand, and a conjecture of I. Ruzsa [@rusza; @perellizannier; @zannierruzsa; @christol], on the other hand. In [@prelim] (3.2 in *loc. cit.*) we proved a weak variant of this conjecture with the purpose of applying it to a case of a generalization of the “Hadamard quotient theorem” to higher genus and positive equicharacteristic (Theorem 1.7 in [*loc. cit.*]{}). In the present note we prove another weak variant, and apply it to obtain a lower bound on the growth of an orbit under a regular iteration of affine space.

[**1.2.** ]{} We need to settle some notation before we can state our result. In what follows we denote by $h(\cdot) : \bar{{\mathbb{Q}}}^{\times} \to {\mathbb{R}}^{\geq 0}$ the absolute logarithmic Weil height, whose definition we recall next. For $p$ a finite rational prime let $|\cdot|_p$ by the $p$-adic absolute value on ${\mathbb{Q}}$ normalized by $|p|_{p} = 1/p$, while for $p = \infty$ we consider the ordinary (archimedean) absolute value. For $v$ a place a number field $K$ lying over the place $p$ of ${\mathbb{Q}}$, and for $x \in K$, let $|x|_{v} := |N_{K_v/{\mathbb{Q}}_p}(x)|_p^{1/[K:{\mathbb{Q}}]}$. For $\mathbf{x} = [x_0:x_1:\cdots:x_r] \in {\mathbb{P}}^r(K)$ we then have $$\begin{aligned}
h(\mathbf{x}) := \sum_v \max_j \log{|x_j|_v}.\end{aligned}$$ This definition is independent of the choice of the number field $K$. Viewing ${\mathbb{A}}^r(\bar{{\mathbb{Q}}})$ as the affine piece $[1:x_1:\ldots:x_r]$ in ${\mathbb{P}}^r(\bar{{\mathbb{Q}}})$, we consider the affine height $h_{\mathrm{aff}}(\alpha_1,\ldots,\alpha_r) := h([1:\alpha_1:\cdots:\alpha_r])$. Finally, we write as usual $h(\alpha) = h_{\mathrm{aff}}(\alpha)$ for $\alpha \in \bar{{\mathbb{Q}}} = {\mathbb{A}}^1(\bar{{\mathbb{Q}}})$.

For a polynomial $F \in K[\mathbf{x}]$ in several variables over $K$, we write $h(F)$ for the height of its set of coefficients, viewed as a point in a projective space. For $F = \sum_{n \geq 0} a_nt^n \in K[[t]]$, let $F_{/N} := \sum_{n=0}^N a_nt^n \in K[t]$ be the polynomial truncation modulo $t^{N+1}$.

[**1.3.** ]{} To motivate our result we make the following trivial observation. If a point $P \in {\mathbb{Z}}^r$ has infinite orbit under a set-theoretic mapping $f: {\mathbb{Z}}^r \to {\mathbb{Z}}^r$, then $H_n := \max_{0 \leq i \leq n} \exp(h(f^iP))$ satisfies $(2H_n+1)^r > n$, whence, in the limit, $$\limsup_{n \in {\mathbb{N}}} \frac{h_{\mathrm{aff}}(f^nP)}{\log{n}} \geq 1/r.$$ The result of this note is that when the mapping $f$ has an algebraic structure, and outside of a degenerate situation, this trivial inequality can be improved exponentially.

[**Theorem 1.4.** ]{} [*Consider regular maps $f : {\mathbb{A}}_{\bar{{\mathbb{Q}}}}^r \to {\mathbb{A}}_{\bar{{\mathbb{Q}}}}^r$ and $\lambda : {\mathbb{A}}_{\bar{{\mathbb{Q}}}}^r \to {\mathbb{A}}_{\bar{{\mathbb{Q}}}}^1$. Let $P \in {\mathbb{A}}^r(\bar{{\mathbb{Q}}})$ be a point such that $$\limsup_{n \in {\mathbb{N}}} \frac{\log{h(\lambda(f^nP))}}{\log{n}} < 1/r.$$ Then there is a $d \in {\mathbb{N}}$ and polynomials $p_0,\ldots,p_{d-1} \in \bar{{\mathbb{Q}}}[x]$ such that $\lambda(f^{nd+i}P) = p_i(n)$ for all $i = 0, \ldots,d-1$ and $n \gg 0$.*]{}

The following is an immediate corollary, obtained by taking $\lambda$ to be the coordinate projections.

[**Corollary 1.5.** ]{} [*In the setup of Theorem 1.3, either the Zariski closure of the orbit $(f^nP)_{n \in {\mathbb{N}}}$ is a union of rational curves in ${\mathbb{A}}^r$, or else $$\limsup_{n \in {\mathbb{N}}} \frac{\log{h_{\mathrm{aff}}(f^nP)}}{\log{n}} \geq 1/r.$$*]{}

Another corollary arises in taking $f$ to be a recurrence of the form $$(x_1,\ldots,x_{r-1},x_r) \mapsto (x_2,\ldots,x_r,p(x_1,\ldots,x_r))$$ and $\lambda$ the projection onto the last coordinate.

[**Corollary 1.6.** ]{} [*Consider a sequence $A : {\mathbb{N}}\to {\mathbb{Z}}$ satisfying a finite polynomial recurrence $A(n+r) = p(A(n+r-1),\ldots,A(n))$, where $p \in {\mathbb{Z}}[T_1,\ldots,T_r]$. Then either $$\limsup_{n \in {\mathbb{N}}} \frac{\log\log{|A(n)|}}{\log{n}} \geq 1/r,$$ or else there is a $d \in {\mathbb{N}}$ such that each of the sequences $(A(nd+i))_{n \geq 0}$, $0 \leq i < d$, is a polynomial for $n \gg 0$.*]{}

We will derive Theorem 1.4 from a criterion for the rationality of a formal function on the projective line, which we formulate and prove in the next section. Results of this type date back to the short paper [@zannierposk] by U. Zannier.

[**2. A rationality criterion**]{}

In what follows, $K$ is a number field and $S$ a finite set of places of $K$ including all archimedean places. The following is a particular case of the conjecture formulated in the appendix to [@prelim].

[**Conjecture 2.1.** ]{} [*Consider $F \in O_{K,S}[[t]]$ a formal power series with $S$-integral coefficients which has a positive radius of convergence $R_v > 0$ at each place $v$ of $K$. For each prime (closed point) $s \in  {\mathrm{Spec} \,}{O_K}$ let $h_s \in {\mathbb{N}}_0 \cup \{\infty\}$ be $+\infty$ if either $s \in S$ or the reduction $F \mod{s} \in k(s)[[t]]$ is not in $k(s)(t)$; and the degree of the rational function $F\mod{s}$ otherwise. If $$\label{shp}
\sum_{v} \log{R_v} + \liminf_{n \in {\mathbb{N}}} \Big\{ \frac{1}{n} \frac{1}{[K:{\mathbb{Q}}]} \sum_{s : \,h_s < n} \log{|k(s)|} \Big\} > 0,$$ then the power series $F \in K(t)$ is rational.*]{}

This conjecture is sharp, in the sense that there are uncountably many power series $F$ for which the quantity on the left-hand side of (\[shp\]) is zero. It extends an old conjecture of I. Ruzsa [@rusza], and appears intractable to our current techniques. In this section we prove the following crude variant.

[**Proposition 2.2.** ]{} [*Let $\eta > 0$. In the setup of Conjecture 2.1, assume instead that the inequality $$\label{strongeq}
  \frac{1}{[K:{\mathbb{Q}}]}  \sum_{s \, : \, h_s < n/(2+\eta)} \log{|k(s)|}  > \frac{3}{2}\log{n} + \Big( 1+\frac{1}{\eta} \Big) h(F_{/n}) + \frac{1}{2}\log{|D_{K/{\mathbb{Q}}}|}$$ holds for all $n \gg 0$. Then $F \in K(t)$ is rational.* ]{}

[*Proof.* ]{} The proof is a variant of that presented to the algebraicity criterion 3.2 in [@prelim]. Without loss of generality we may assume $\eta$ to be rational. Letting $L$ be a large integer parameter such that $\eta L \in {\mathbb{Z}}$, Siegel’s lemma (see 3.1 in [@prelim] and the references therein) for $M := L$ equations in $N := (1+\eta) L$ unknowns yields polynomials $P \in K[t]$ and $Q \in K[t] \setminus \{0\}$ of degrees less than $(1+\eta)L$ such that $$Q(t)F(t) - P(t) \equiv 0 \mod{t^{ (2+\eta)L }}$$ and $$\begin{aligned}
 \label{estimate}
h(Q) \leq \frac{1}{2}\log{N} + \frac{1}{2}\log{|D_{K/{\mathbb{Q}}}|} + \frac{1}{\eta} h\big( F_{/(2+\eta)L} \big).\end{aligned}$$ It follows from (\[strongeq\]) and (\[estimate\]) that there is an $L < \infty$ such that all $n \geq (2+\eta)L$ satisfy $$\label{contr}
\frac{1}{[K:{\mathbb{Q}}]} \sum_{s : \, h_s < n/(2+\eta)} \log{|k(s)|} > \log{N} + h(Q) + h(F_{/n}).$$ [*We claim that then $F = P/Q$, hence $F$ is rational.*]{} Assuming otherwise, let $n$ be the minimum integer such that $Q(t)F(t) - P(t) \not\equiv 0 \mod{t^n}$; by construction, $n > (2+\eta)L$. Consider a prime $s$ of $K$ with $h_s < n/(2+\eta)$, and write $F_s := A_s/B_s$ the reduction at $s$, with $A_s, B_s \in k(s)[t]$, $\deg{A_s}, \deg{B_s} < n/(2+\eta)$. Then, denoting by a tilde the reduction at $s$, we have $A_s(t)\widetilde{Q}(t) - B_s(t)\widetilde{P}(t) \equiv 0 \mod{t^{n-1}}$. The degree of this polynomial is less than $(1+\eta)L + n/(2+\eta)$, which by our assumption $n > (2+\eta)L$ does not exceed $n-1$. It follows that the polynomial is identically zero, hence the coefficients of $Q(t)F(t) - P(t)$ all reduce to zero at $s$.

On the other hand, we have assumed that $t^n$ appears with a non-zero coefficient $c \in K \setminus \{0\}$ in $Q(t)F(t) - P(t)$. (This is just the coefficient of $t^n$ in $Q(t)F(t)$.) Thus the product formula yields $$\label{productform}
 \sum_{\textrm{all } v} -\log{|c|_v} = 0.$$ At the places corresponding to the primes $s$ with $h_s < n/(2+\eta)$, the previous paragraph shows that the contribution to (\[productform\]) is at least $\frac{1}{[K:{\mathbb{Q}}]}\log{|k(s)|}$. The absolute value of the sum of the remaining contributions does not exceed $h(c)$, and (\[productform\]) yields the lower bound $$\label{lower}
 h(c) \geq \frac{1}{[K:{\mathbb{Q}}]} \sum_{s : \, h_s < n/(2+\eta)} \log{|k(s)|}.$$ To estimate $h(c)$ from above, we use the easy bound $$h(\alpha_1 + \cdots + \alpha_r) \leq \log{r} + \sum_v \max_j \log^+{|\alpha_j|_v},$$ applied to the sum defining $c$ as the coefficient of $t^n$ in $Q(t)F(t)$. We obtain: $$\label{upper}
h(c) \leq \log{N} + h(Q) + h(F_{/n}).$$ Taken together with (\[lower\]) this contradicts (\[contr\]), thus forcing $F = P/Q$ as claimed.

[**3. Proof of Theorem 1.4** ]{}

There is a number field $K$ and a finite set $S$ of its places, including all archimedean places, such that the triple $(f,\lambda,P)$ has a model over $O_{K,S}$. We will apply Proposition 2.2 to the formal power series $\Phi := \sum_{n \geq 0} \lambda(f^nP) t^n \in O_{K,S}[[t]]$. If the power series $\Phi$ is rational, the conclusion of Theorem 1.4 follows from the explicit descriptions of coefficients of rational power series as confluent power sums. If $\Phi$ is not rational, Proposition 2.2 with $\eta := 1$ implies the lower bound inequality $$\label{reverse}
 h(\Phi_{/n}) \geq  \frac{1}{2[K:{\mathbb{Q}}]}  \sum_{s \, : \, h_s(\Phi) < n/3} \log{|k(s)|} - O(\log{n})$$ for infinitely many $n \in {\mathbb{N}}$. On the other hand, for $s \notin S$ a prime of $K$, the iteration $f : {\mathbb{A}}_{O_{K,S}}^r \to {\mathbb{A}}_{O_{K,S}}^r$ reduces mod $s$ to an iteration $f\mod{s} : {\mathbb{A}}_{k(s)}^r \to {\mathbb{A}}_{k(s)}^r$ over a set with $|k(s)|^r$ elements, hence $h_s(\Phi) \leq 2|k(s)|^r$. Consequently, by the prime number theorem, (\[reverse\]) yields $$\begin{aligned}
 \label{ataka}
h(\Phi_{/n}) \geq  \frac{1}{3}  (n/6)^{1/r} \quad \textrm{for arbitrarily large } n \in {\mathbb{N}}.
 \end{aligned}$$ We have $h(\Phi_{/n}) \leq |S|\max_{j \leq n} h(\lambda(f^jP))$, and the conclusion of the theorem follows from (\[ataka\]).

[**4. Two conjectures**]{}

We end this note by recording two conjectures related to the setup of Theorem 1.4. Problems of this type have been posed by J.H. Silverman in [@silvrational].

[**Conjecture 4.1.** ]{} [*Let $X$ be a complex projective variety, $f : X \dashrightarrow X$ a rational self-map, $\lambda : X \dashrightarrow {\mathbb{P}}^1$ a non-constant rational function, and $P \in X({\mathbb{C}})$ a point with well-defined forward orbit. Then the set $\{ n \mid \lambda(f^nP) = 0 \} \subset {\mathbb{N}}_0$ is the union of a finite set with finitely many full arithmetic progressions.* ]{}

[**Conjecture 4.2.** ]{} [*Let $A/\bar{{\mathbb{Q}}}$ be an abelian variety and $\lambda : A \dashrightarrow {\mathbb{P}}^1$ a non-constant rational function. Consider a point $P \in A(\bar{{\mathbb{Q}}})$. If $h\big( \lambda([n]P) \big) = o(n^2)$ then there is a surjective homomorphism $A \twoheadrightarrow B$ to an abelian variety mapping $P$ to a torsion point.*]{}

[99]{}

Amice Y.: Les nombres $p$-adiques, [*Collection SUP: Le Mathématicien*]{}, vol. [**14**]{}, Presses Universitaires de France, Paris, 1975.

Bombieri E., W. Gubler: Heights in Diophantine Geometry, [*Cambridge New Mathematical Monographs*]{} (2006).

Bombieri E., J. Vaaler: On Siegel’s lemma, [*Invent. Math.*]{} [**73**]{} (1983), pp. 11–32.

Christol G.: Globally bounded solutions of differential equations, in [*Analytic Number Theory Proc. Jap., Fr. Symp. Tokyo, 1988*]{}, Lecture Notes in Math. vol. [**1434**]{} (1990), pp. 45–64.

Dimitrov V.: A note on a generalization of the Hadamard quotient theorem. (Preprint.) `arXiv:1309.1920`

Perelli A., U. Zannier: On recurrent mod $p$ sequences, [*J. reine Angew. Math.*]{}, vol. [**348**]{} (1984), pp. 135–146.

Ruzsa I.: On congruence preserving functions (in Hungarian), [*Mat. Lapok*]{} [**22**]{} (1971), pp. 125–134.

Silverman, J. H.: Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space, [*Ergodic Theory and Dynamical Systems*]{}, Available on CJO 2012 doi:10.1017/etds.2012.144. `arXiv:1111.5664v2`.

Zannier U.: A note on recurrent mod $p$ sequences, [*Acta Arithmetica*]{}, vol. [**41**]{} (1982), pp. 277–280.

Zannier U.: On periodic mod $p$ sequences and $G$-functions (On a conjecture of Ruzsa), [*Manuscripta Math.*]{}, vol. [**90**]{} (1996), pp. 391–402.
