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A note on polynomial equidistribution and recurrence in finite characteristic

Ethan Ackelsberg, Vitaly Bergelson

Abstract

This paper addresses the topic of equidistribution and recurrence for polynomial sequences over function fields. The main focus is to note and correct two small errors in [V. Bergelson and A. Leibman, A Weyl-type equidistribution theorem in finite characteristic, Adv. Math. 289 (2016) 928-950], contextualized within the broader developing literature on number theory and additive combinatorics in function fields. Connected with the resolution of these issues, we also prove new results characterizing intersective polynomials in finite characteristic in terms of various algebraic, combinatorial, and dynamical properties.

A note on polynomial equidistribution and recurrence in finite characteristic

Abstract

This paper addresses the topic of equidistribution and recurrence for polynomial sequences over function fields. The main focus is to note and correct two small errors in [V. Bergelson and A. Leibman, A Weyl-type equidistribution theorem in finite characteristic, Adv. Math. 289 (2016) 928-950], contextualized within the broader developing literature on number theory and additive combinatorics in function fields. Connected with the resolution of these issues, we also prove new results characterizing intersective polynomials in finite characteristic in terms of various algebraic, combinatorial, and dynamical properties.
Paper Structure (14 sections, 8 theorems, 26 equations, 2 figures)

This paper contains 14 sections, 8 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Basic algebraic objects in finite characteristic
  4. Well distribution
  5. Polynomial sequences
  6. Subtori
  7. Equidistribution theorem
  8. Corrected formulations of the equidistribution theorem
  9. A note on the counterexample of Champagne--Ge--Lê--Liu--Wooley
  10. Impact on subsequent papers
  11. Intersective polynomials
  12. Property (P1)
  13. Proof of equivalences
  14. Additional consequences of intersectivity

Key Result

Theorem 3.1

Any additive polynomial $\mathcal{Z}$-sequence $\eta(n)$ in $\mathcal{T}^c$ is well distributed in a set of the form $\mathcal{F}(\eta) + \eta(K)$, where $\mathcal{F}(\eta)$ is a $\Phi$-subtorus of level $\le \log_p \deg{\eta}$ of $\mathcal{T}^c$ and $K$ is a finite subset of $\mathcal{Z}$. For any

Figures (2)

  • Figure 1: Relations between combinatorial and recurrence properties in Definition \ref{['definitions']}.
  • Figure 2: Implications demonstrated in the proof.

Theorems & Definitions (18)

  • Theorem 3.1
  • Remark 3.2
  • Example 3.3
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • Proposition 3.7
  • proof
  • Proposition 4.1
  • ...and 8 more