Pointwise ergodic theorems for non-conventional bilinear averages along $(\lfloor n^c\rfloor,-\lfloor n^c\rfloor)$

Abstract

For every $c\in(1,23/22)$ and every probability dynamical system $(X,\mathcal{B},\mu,T)$ we prove that for any $f,g\in L^{\infty}_{\mu}(X)$ the bilinear ergodic averages \[ \frac{1}{N}\sum_{n=1}^Nf(T^{\lfloor n^c\rfloor}x)g(T^{-\lfloor n^c\rfloor}x)\quad\text{converge for $\mu$-a.e. $x\in X$.} \] In fact, we consider more general sparse orbits $(\lfloor h(n)\rfloor,-\lfloor h(n)\rfloor)_{n\in\mathbb{N}}$, where $h$ belongs to the class of the so-called $c$-regularly varying functions. This is the first pointwise result for bilinear ergodic averages taken along deterministic sparse orbits where modulation invariance is present.