On the multiplicative group generated by two primes in $\mathbf{Z}/Q\mathbf{Z}$

TL;DR

This paper investigates the behavior of the multiplicative group generated by two primes in modular arithmetic, focusing on return times to specific sets and revisiting bounds on gcd of S-unit differences using the S-adic subspace theorem.

Contribution

It introduces a new analysis of the multiplicative group generated by two primes in modular settings and applies the S-adic subspace theorem to bound gcd of S-unit differences.

Findings

Characterization of return times to certain sets in modular arithmetic

Bounds on gcd of S-unit differences derived using the S-adic subspace theorem

Insights into the structure of groups generated by two primes in residue classes

Abstract

We study the action of the multiplicative group generated by two prime numbers in $\mathbf{Z}/Q\mathbf{Z}$. More specifically, we study returns to the set $([-Q^\varepsilon,Q^\varepsilon]\cap \mathbf{Z})/Q\mathbf{Z}$. This is intimately related to the problem of bounding the greatest common divisor of $S$-unit differences, which we revisit. Our main tool is the $S$-adic subspace theorem.