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Efficient equidistribution of periodic nilsequences and applications

James Leng

Abstract

This is a companion paper to arXiv:2312.10772. We deduce an equidistribution theorem for periodic nilsequences and use this theorem to give two applications in arithmetic combinatorics. The first application is quasi-polynomial bounds for a certain complexity one polynomial progression, improving the iterated logarithm bound previusly obtained. The second application is a proof of the quasi-polynomial $U^4[N]$ inverse theorem. In work with Sah and Sawhney, we obtain improved bounds for sets lacking nontrivial $5$-term arithmetic progressions.

Efficient equidistribution of periodic nilsequences and applications

Abstract

This is a companion paper to arXiv:2312.10772. We deduce an equidistribution theorem for periodic nilsequences and use this theorem to give two applications in arithmetic combinatorics. The first application is quasi-polynomial bounds for a certain complexity one polynomial progression, improving the iterated logarithm bound previusly obtained. The second application is a proof of the quasi-polynomial inverse theorem. In work with Sah and Sawhney, we obtain improved bounds for sets lacking nontrivial -term arithmetic progressions.
Paper Structure (33 sections, 45 theorems, 371 equations)

This paper contains 33 sections, 45 theorems, 371 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Discussion on the proof of the $U^4(\mathbb{Z}/N\mathbb{Z})$ inverse theorem
  4. Organization of the paper
  5. Acknowledgements
  6. Notation
  7. Bracket polynomial heuristics
  8. Proof of \ref{['twostepbracket']}
  9. The refined bracket polynomial lemma
  10. Finishing the proof
  11. The degree two nilsequence case
  12. The two-step polynomial sequence case
  13. The general periodic case
  14. The complexity one polynomial Szemerédi theorem
  15. Proof of \ref{['polynomialinverse']}
  16. ...and 18 more sections

Key Result

Theorem 1

Let $G/\Gamma$ is a nilmanifold and $F(g(n)\Gamma)$ is a nilsequence with $F$ a $G_{(s)}$-vertical character with nonzero frequency $\xi$. Suppose Then $F$ is "morally" a step $\le s - 1$ nilsequence with "good bounds."

Theorems & Definitions (92)

  • Theorem 1: Informal Equidistribution Theorem
  • Theorem 2
  • Remark 1.1
  • Theorem 3
  • Theorem 4
  • Definition 2.1: Periodic nilsequences
  • Definition 2.2: Smoothness norms
  • Definition 3.1: Elementary two-step nilmanifold
  • Definition 3.2: Elementary bracket quadratic
  • Definition 3.3: Bilinear form of an elementary two-step nilmanifold
  • ...and 82 more