Rational numbers in $\times b$-invariant sets

Abstract

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