Pointwise convergence of double ergodic averages along certain non-polynomial sequences

Abstract

Fix $c\in (1,23/22)$. Let $\alpha$ and $\beta$ be two distinct non-zero real numbers with $|\alpha|\neq |\beta|$. It is shown that for any measure preserving system $(X,\mathcal{X},\mu,T)$ and any $f,g\in L^{\infty}(\mu)$, the limit \begin{equation*} \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(T^{\lfloor \alpha n^c \rfloor}x)g(T^{\lfloor \beta n^c \rfloor}x) \end{equation*} exists for $\mu$-a.e. $x\in X$. Meanwhile, a multidimensional version of the above result is also presented.