Group topologies on integers and S-unit equations

TL;DR

The paper investigates group topologies on integers where specific sequences, including exponential sums, converge to zero, using number theory to solve an open problem about a particular sequence being a T-sequence.

Contribution

It constructs new Hausdorff group topologies on integers for sequences related to S-integers and solves an open problem about the sequence 2^n + 3^n being a T-sequence.

Findings

Constructed topologies where S-integers converge to zero

Confirmed 2^n + 3^n is a T-sequence

Linked number theory with topological group properties

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.