Take the tour ×  Is such a thing even possible?  There's not
much to say really. Obviously if there was a set it would be
full of transcendental numbers. This led me to think of a
function generating transcendental numbers (given a
transcendental number) closed under addition and multiplication
- an easy one is algebraic function but these are clearly
countable however. (Even worse the numbers would have to be
algebraically independent which means you have to cut back on
the algebraic functions.).  Any insights, guys?  share|improve
this question I think it would either have full measure or
measure $0$ if it exists. Call the set in question $S$, and take
some $r\in S$. Then $S\mod r = [0,r]\cap S$ is an uncountable
set of irrationals contained in the unit interval (since $s\in
S,n\in\mathbb{Z}\implies s+nr\in S$). The transformation given
by adding $q\in S$ not a rational multiple of $r$ is ergodic but
fixes $[0,r]\cap S$ so $[0,r]\cap S$ must have full measure or
measure $0$, and this can be extended to $S$