Rational numbers in $\times b$-invariant sets

TL;DR

This paper investigates the structure of certain invariant sets under the multiplication-by-b operation, proving finiteness of specific rational numbers within these sets and analyzing the prime divisors of their denominators.

Contribution

It establishes the finiteness of rational numbers with restricted prime divisors in non-dense, -invariant sets and provides quantitative bounds on their denominators.

Findings

Finiteness of rational numbers with denominators divisible only by primes in S.

Quantitative bounds on the largest prime divisors of denominators.

Invariance under  operation constrains rational numbers in the set.

Abstract

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 $A \subset [0,1)$ such that $A \cap \mathbb{Q}$ 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.