A structure theorem for polynomial return-time sets in minimal systems

TL;DR

This paper characterizes the structure of polynomial return-time sets in minimal systems, linking them to maximal pronilfactors, and applies this to establish new recurrence theorems and relate two conjectures.

Contribution

It provides a structure theorem connecting return-time sets with pronilfactors in minimal systems, advancing understanding of polynomial recurrence.

Findings

Return-time sets coincide with those in maximal infinite-step pronilfactors.

Established new multiple recurrence theorems for linear and polynomial cases.

Proved the equivalence of two previously separate conjectures.

Abstract

We investigate the structure of return-time sets determined by orbits along polynomial tuples in minimal topological dynamical systems. Building on the topological characteristic factor theory of Glasner, Huang, Shao, Weiss, and Ye, we prove a structure theorem showing that, in a minimal system, return-time sets coincide -- up to a non-piecewise syndetic set -- with those in its maximal infinite-step pronilfactor. As applications, we establish three new multiple recurrence theorems concerning linear recurrence along dynamically defined syndetic sets and polynomial recurrence along arithmetic progressions in minimal and totally minimal systems. We also show how our main theorem can be used to prove that two previously separate conjectures -- one due to Glasner, Huang, Shao, Weiss, and Ye and the other due to Leibman -- are equivalent.