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Double Recurrence and Almost Sure Convergence: Primes and Weighted Theory

Jan Fornal, Ben Krause

Abstract

Let $(X,μ)$ be a probability space equipped with an invertible, measure-preserving transformation $T\colon X \to X$. We exhibit a wide class of weights $w$ so that whenever $f,g \in L^{\infty}(X)$, the bilinear ergodic averages \[ \frac{1}{N} \sum_{n \leq N} w(n)\, T^{an}f \cdot T^{bn}g, \qquad a,b \in \mathbb{Z} \] converge $μ$-almost surely. This class encompasses the von Mangoldt function, resolving Problem 12 from Frantzikinakis' survey on open problems in ergodic theory, the divisor function, the sum-of-two-squares representation function, etc., as well as their restrictions to lower-density Piatetski-Shapiro sequences of the form $\{\lfloor k^{c}\rfloor : k \in \mathbb{N}\}$, $1 \leq c < 7/6$. Our methods combine combinatorial number theory and higher-order Fourier analysis with classical Fourier-analytic/martingale-based methods; the role of $U^{3}$ analysis is particularly significant.

Double Recurrence and Almost Sure Convergence: Primes and Weighted Theory

Abstract

Let be a probability space equipped with an invertible, measure-preserving transformation . We exhibit a wide class of weights so that whenever , the bilinear ergodic averages converge -almost surely. This class encompasses the von Mangoldt function, resolving Problem 12 from Frantzikinakis' survey on open problems in ergodic theory, the divisor function, the sum-of-two-squares representation function, etc., as well as their restrictions to lower-density Piatetski-Shapiro sequences of the form , . Our methods combine combinatorial number theory and higher-order Fourier analysis with classical Fourier-analytic/martingale-based methods; the role of analysis is particularly significant.
Paper Structure (46 sections, 55 theorems, 744 equations)

This paper contains 46 sections, 55 theorems, 744 equations.

Table of Contents

  1. Introduction
  2. History
  3. Proof Overview
  4. Structure
  5. Open Problems
  6. Acknowledgments
  7. Preliminaries
  8. General Notation
  9. Asymptotic Notation
  10. Fourier Transforms
  11. Gowers Norms and Technology
  12. Quantifying Convergence
  13. Organizing Parameters
  14. Properties and Examples of Admissible Weights
  15. Properties of Admissible Weights
  16. ...and 31 more sections

Key Result

Theorem 1.1

Let $(X,\mu,T)$ be a measure-preserving system, and suppose that $T_1,T_2$ are powers of $T$.Throughout, we allow our powers to be negative, i.e. to involve the transformation $T^{-1}$. Then for $f,g \in L^\infty(X)$, the bilinear ergodic averages converge $\mu$-almost surely.

Theorems & Definitions (89)

  • Theorem 1.1: Bourgain's Double Recurrence Theorem
  • Theorem 1.2: Special Case
  • Definition 1.8
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 1.5
  • Theorem : Pointwise Ergodic Theorem
  • Theorem : Wiener-Wintner Ergodic Theorem
  • Theorem : Return Times Theorem
  • Theorem : Bourgain's Polynomial Ergodic Theorem
  • ...and 79 more