Rational numbers in \times b-invariant sets.

Let b \geq 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 \subseteq [0,1) which is invariant under \times 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. [ABSTRACT FROM AUTHOR]