Pointwise ergodic theorems along oscillating sequences

Abstract

We establish the first pointwise convergence result for nonconventional ergodic averages modeled along a large class of oscillating sequences, treating, for example, orbits of the form $\big(\lfloor n^d\sin(eπn)\rfloor\big)_{n\in\mathbb{N}}$, $d\in\mathbb{N}$. More specifically, for every probability space $(X,\mathcal{B},μ)$ equipped with an invertible measure-preserving transformation $T\colon X\to X$ and every $f\in L^p_μ(X)$ with $p\in(1,\infty]$, we prove that the limit \[ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nf\big(T^{\lfloor P(n)ψ(αn)\rfloor}x\big)\quad\text{exists for $μ$-a.e. $x\in X$,} \] where $P$ is a non-constant polynomial with real coefficients, $ψ$ is $1$-periodic and real-analytic, and $α$ has irrationality exponent smaller than $1+\sqrt{\operatorname{deg}(P)+1}$. On the one hand, the latter assumption is satisfied by a.e. $α\in\mathbb{R}$, and, on the other hand, our result covers any of the following concrete orbits \[ \lfloor n\cos(eπn)\rfloor\text{,}\quad\lfloor n^{37}\sin(n)\rfloor\text{,}\quad\big\lfloor \sqrt{2}n^{10}e^{\sin(π(\sqrt[5]{7}+1) n)}\big\rfloor\text{,}\quad \big\lfloor \big((\log 2) n^3+e n\big)e^{e^{\cos(π\sqrt{2} n)}}\big\rfloor\text{.} \] The proof employs the so-called one-frequency circle method for estimating exponential sums with phases $\big(ξ\lfloor P(n)ψ(αn)\rfloor\big)_{1\le n\le N}$, but the major arc approximants of our exponential sums do not consist of their direct continuous counterparts. In the major arc regime we decouple the $ψ$-oscillatory part of the phase from the polynomial one, and approximate the exponential sum by $\int_0^1\big(\frac{1}{N}\int_0^Ne^{2πiξP(t)ψ( s)}dt\big)ds$. This represents a novel phenomenon for problems of this kind, and, interestingly, our proof exploits the coefficient-free variational estimates of Radon-type operators.