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Ergodic theorems for bilinear averages, Roth's Theorem and Corners along fractional powers

Leonidas Daskalakis

TL;DR

This work advances ergodic theory by establishing bilinear averages along fractional powers for two commuting measure-preserving transformations, proving that for c in (1,23/22) the fractional-power averages converge to the classical integer-power limits. It introduces a unifying change-of-variables method that reduces sparse fractional-power orbits to standard averages, via a main-term decomposition and precise $U^3$-norm control of the error; Calderón transference then yields the ergodic theorems in the general setting of $c$-regularly varying orbits. The paper also delivers quantitative Roth-type results and corners theorems for fractional powers, with density bounds matching the state of the art and corollaries for primes, thus addressing open problems in Frantzikinakis’ survey. Overall, the results broaden the scope of multiple ergodic averages to sparse and irregular orbits, providing robust techniques that connect ergodic theory, additive combinatorics, and number theory.

Abstract

We prove that for every $c\in(1,23/22)$, every probability space $(X,\mathcal{B},μ)$ equipped with two commuting measure-preserving transformations $T,S\colon X\to X$ and every $f,g\in L^{\infty}_μ(X)$ we have that the $L^2_μ(X)$-limit \[ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nf(T^{\lfloor n^c\rfloor}x)g(S^{\lfloor n^c\rfloor}x) \] equals the $L^2_μ(X)$-limit $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nf(T^{n}x)g(S^{n}x)$. The approach is based on the author's recently developed technique which may be thought of as a change of variables. We employ it to establish several new results along fractional powers including a Roth-type result for patterns of the form $x,x+\lfloor y^c \rfloor,x+2\lfloor y^c \rfloor$ as well as its ''corner'' counterpart. The quantitative nature of the former result allows us to recover the analogous one in the primes. Our considerations give partial answers to Problem 29 and Problem 30 from Frantzikinakis' open problems survey on multiple ergodic averages. Notably, we cover more general sparse orbits $(\lfloor h(n)\rfloor)_{n\in\mathbb{N}}$, where $h$ belongs to the class of the so-called $c$-regularly varying functions, addressing for example even the orbit $(\lfloor n\log n\rfloor)_{n\in\mathbb{N}}$.

Ergodic theorems for bilinear averages, Roth's Theorem and Corners along fractional powers

TL;DR

This work advances ergodic theory by establishing bilinear averages along fractional powers for two commuting measure-preserving transformations, proving that for c in (1,23/22) the fractional-power averages converge to the classical integer-power limits. It introduces a unifying change-of-variables method that reduces sparse fractional-power orbits to standard averages, via a main-term decomposition and precise -norm control of the error; Calderón transference then yields the ergodic theorems in the general setting of -regularly varying orbits. The paper also delivers quantitative Roth-type results and corners theorems for fractional powers, with density bounds matching the state of the art and corollaries for primes, thus addressing open problems in Frantzikinakis’ survey. Overall, the results broaden the scope of multiple ergodic averages to sparse and irregular orbits, providing robust techniques that connect ergodic theory, additive combinatorics, and number theory.

Abstract

We prove that for every , every probability space equipped with two commuting measure-preserving transformations and every we have that the -limit equals the -limit . The approach is based on the author's recently developed technique which may be thought of as a change of variables. We employ it to establish several new results along fractional powers including a Roth-type result for patterns of the form as well as its ''corner'' counterpart. The quantitative nature of the former result allows us to recover the analogous one in the primes. Our considerations give partial answers to Problem 29 and Problem 30 from Frantzikinakis' open problems survey on multiple ergodic averages. Notably, we cover more general sparse orbits , where belongs to the class of the so-called -regularly varying functions, addressing for example even the orbit .

Paper Structure

This paper contains 9 sections, 18 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Strategy
  3. Notation
  4. Main term extraction
  5. Gowers norm bounds
  6. Bounds by the $U^3$-norm
  7. Bounds for the $U^3$-norm of the error kernel
  8. Calderón's Transference Principle and the proofs of the Ergodic Theorems
  9. Roth's theorem and Corners for fractional powers

Key Result

Theorem 1.1

Assume $c\in(1,23/22)$, and let $(X,\mathcal{B},\mu)$ be a probability space and $T,S$ measure-preserving transformations on $X$ which commute. Then for every $f,g\in L^{\infty}_{\mu}(X)$ we have that the $L^2_{\mu}(X)$-limit equals the $L^2_{\mu}(X)$-limit $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nf(T^{n}x)g(S^{n}x)$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Definition 1.3: $c$-regularly varying functions
  • Theorem 1.5
  • Theorem 1.7
  • Corollary 1.9
  • Theorem 1.10
  • Theorem 1.13
  • Proposition 2.1
  • proof
  • Proposition 2.10
  • ...and 25 more