Multiplicative combinatorial properties of return time sets in minimal dynamical systems

TL;DR

This paper explores how minimal dynamical systems' properties influence the multiplicative structure of return time sets, providing evidence that such sets often contain long geometric progressions, addressing a long-standing open problem.

Contribution

It establishes that for residual points in minimal systems, return time sets contain arbitrarily long geometric progressions, advancing understanding of multiplicative structures in dynamical systems.

Findings

Return time sets contain arbitrarily long geometric progressions for residual points.

Under total minimality, return time sets have positive multiplicative upper Banach density.

Results provide evidence supporting that syndetic sets contain long geometric progressions.

Abstract

We investigate the relationship between the dynamical properties of minimal topological dynamical systems and the multiplicative combinatorial properties of return time sets arising from those systems. In particular, we prove that for a residual sets of points in any minimal system, the set of return times to any non-empty, open set contains arbitrarily long geometric progressions. Under the separate assumptions of total minimality and distality, we prove that return time sets have positive multiplicative upper Banach density along $\mathbb{N}$ and along multiplicative subsemigroups of $\mathbb{N}$, respectively. The primary motivation for this work is the long-standing open question of whether or not syndetic subsets of the positive integers contain arbitrarily long geometric progressions; our main result is some evidence for an affirmative answer to this question.