Rational numbers in ×𝑏-invariant sets

Abstract

Let b ≥ 2 b \geq 2 be an integer and S S be a finite non-empty set of primes not containing divisors of b b . For any non-dense set A ⊆ [ 0 , 1 ) A \subseteq [0,1) which is invariant under × b \times b operation, we prove the finiteness of rational numbers in A A whose denominators can only be divided by primes in S S . A quantitative result on the largest prime divisors of the denominators of rational numbers in A A is also obtained.