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

Abstract

We prove that for every $c\in(1,23/22)$, every probability space $(X,\mathcal{B},\mu)$ equipped with two commuting measure-preserving transformations $T,S\colon X\to X$ and every $f,g\in L^{\infty}_{\mu}(X)$ we have that the $L^2_{\mu}(X)$-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 $L^2_{\mu}(X)$-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 $(\lfloor n\log n\rfloor)_{n\in\mathbb{N}}$.