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

Daniel Glasscock, Andreas Koutsogiannis, Anh N. Le, Joel Moreira, Florian K. Richter, Donald Robertson

TL;DR

This work develops a structure theorem for polynomial return-time sets in minimal topological dynamical systems, showing that return-time behavior along polynomial tuples is governed by the maximal $\infty$-step pronilfactor up to a non-piecewise-syndetic discrepancy. Building on topological characteristic-factor theory and IP-polynomial Szemerédi methods, the authors lift recurrence information from pronilfactors to the ambient system, enabling four main applications: linear recurrence on dynamically syndetic sets, polynomial recurrence along progressions, recurrence in totally minimal systems, and a unification of two major conjectures (Glasner–Huang–Shao–Weiss–Ye and Leibman). The results reduce polynomial recurrence questions to inverse limits of nilsystems, providing new recurrence theorems and a framework to compare conjectural landscapes across topological and ergodic settings. The paper also discusses essential distinctness, interiors, non-invertible extensions, and potential ergodic analogues, outlining several open questions and future directions.

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.

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

TL;DR

This work develops a structure theorem for polynomial return-time sets in minimal topological dynamical systems, showing that return-time behavior along polynomial tuples is governed by the maximal -step pronilfactor up to a non-piecewise-syndetic discrepancy. Building on topological characteristic-factor theory and IP-polynomial Szemerédi methods, the authors lift recurrence information from pronilfactors to the ambient system, enabling four main applications: linear recurrence on dynamically syndetic sets, polynomial recurrence along progressions, recurrence in totally minimal systems, and a unification of two major conjectures (Glasner–Huang–Shao–Weiss–Ye and Leibman). The results reduce polynomial recurrence questions to inverse limits of nilsystems, providing new recurrence theorems and a framework to compare conjectural landscapes across topological and ergodic settings. The paper also discusses essential distinctness, interiors, non-invertible extensions, and potential ergodic analogues, outlining several open questions and future directions.

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.

Paper Structure

This paper contains 35 sections, 40 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Applications
  4. Linear recurrence along dynamically syndetic sets
  5. Polynomial recurrence along arithmetic progressions
  6. Recurrence in totally minimal systems
  7. Conjectures of Glasner-Huang-Shao-Weiss-Ye and Leibman
  8. Outline
  9. Acknowledgments
  10. Preliminaries
  11. Families of subsets of Z
  12. Topological dynamics
  13. Nilsystems and pro-nilsystems
  14. A structure theorem for return-time sets
  15. Basics
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $(X,T)$ be a minimal and invertible topological dynamical system. Denote by $(X_\infty,T)$ its maximal $\infty$-step pro-nilfactor, and let $\pi\colon X\to X_\infty$ be the associated factor map. For all $d \in \mathbb{N}$, all nonempty, open $U_1, \ldots, U_d \subseteq X$, and all essentially d

Theorems & Definitions (84)

  • Theorem 1.1: cf. Qiu23 and \ref{['thm_qiu_equivalent']}
  • Theorem 1
  • Theorem 1.2: glasner_huang_shao_weiss_ye_2020
  • Remark 1.3
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Conjecture 1.4: glasner_huang_shao_weiss_ye_2020
  • Conjecture 1.5: See Leibman10b
  • Theorem 5
  • ...and 74 more