Group topologies on integers and S-unit equations

Abstract

A sequence of integers $\{s_n{\}}_{n\in \mathbb{N}}$ is called a T-sequence if there exists a Hausdorff group topology on $\mathbb{Z}$ such that $\{s_n{\}}_{n\in \mathbb{N}}$ converges to zero. For every finite set of primes $S$ we build a Hausdorff group topology on $\mathbb{Z}$ such that every growing sequence of $S$-integers converges to zero. As a corollary, we solve in the affirmative an open problem by I.V. Protasov and E.G. Zelenuk asking if $\{2^n+3^n{\}}_{n\in \mathbb{N}}$ is a T-sequence. Our results rely on a nontrivial number-theoretic fact about $S$-unit equations.