Weighted ergodic averages along subpolynomials in Hardy fields and applications

Abstract

We establish new pointwise convergence results for weighted ergodic averages along sequences of the form \( (\lfloor a(n) \rfloor)_{n \in \mathbb{N}}, \) where $a(x)$ is a subpolynomial function in a Hardy field. For example, we establish pointwise convergence of logarithmic averages along sequences of the form $(\lfloor n^k + \log^{c} n \rfloor)_{n \in \mathbb{N}}$, where $k \in \mathbb{N} \cup \{0\}$ and $c > 0$. This result should be juxtaposed with the fact that either for $k=0$ or for $k \geq 2$ and for sufficiently small $c>0$ (depending on $k$), the standard ergodic averages along these sequences fail to converge pointwise. We also obtain pointwise joint ergodicity results for multiple weighted ergodic averages along slow Hardy field functions. For example, it follows from our results that for $c> 0$ and for any $f, g \in L^{\infty} (λ)$, \begin{equation*} \lim_{N \rightarrow \infty} \frac{1}{\log N } \sum_{n=1}^{N} \frac{1}{n} f(T_b^{\lfloor \log^c n \rfloor}x) \, g(T_G^{\lfloor \log^c n \rfloor} x) = \int f \, d λ\cdot \int g \, d μ_G \quad \text{for almost every } x \in [0,1], \end{equation*} where $T_b:[0,1] \rightarrow [0,1]$ is the times-$b$ map defined by $T_b x = bx \, \bmod \, 1 $ and $T_G:[0,1] \rightarrow [0,1]$ is the Gauss map defined by $T_G(x) = \frac{1}{x} \bmod \, 1$ for $x \ne 0$ and $T_G (0) =0$. Here $λ$ is the Lebesgue measure on $[0,1]$ and $μ_G$ is the Gauss measure on $[0,1]$ given by $μ_G (A) = \frac{1}{ \log 2} \int_A \frac{1}{1+x} dx$ for any measurable set $A \subset [0,1]$.