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

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 $([-Q^{\epsilon },Q^{\epsilon }]\cap \mathbf{Z})/Q\mathbf{Z}$. 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.

arXiv comment14 pages, final accepted version, revision based on referee's comments, new title, improved presentation, to appear in Analysis at Large: Dedicated to the Life and Work of Jean Bourgain