Pointwise Ergodic Theorems Along $\lfloor p^c\rfloor$ for Every $c>1$

Abstract

Let $c>1$ be noninteger, and let $Λ$ denote the von Mangoldt function. The averages \[ A_N^{c} f(x):=\frac1N\sum_{n=1}^{N} Λ(n)f\bigl(T^{\lfloor n^c\rfloor}x\bigr) \] satisfy the variational estimate \[ \bigl\|\mathcal{V}^r(A_N^{c} f:N\in\mathbb{N})\bigr\|_{L^p(X)} \lesssim_{c,p,r}\|f\|_{L^p(X)} \] for every $1<p<\infty$ and $r>2$, on every $σ$-finite measure-preserving system. In particular, the normalized averages along fractional powers of primes converge almost everywhere. The main arithmetic ingredient is a uniform bound on the correlations of $Λ-1$ with fractional-power exponential phases, yielding arbitrarily large powers of logarithmic gain. We obtain this estimate through short-interval polynomial approximation of the fractional phase and higher-order discorrelation estimates. The same approach yields almost-everywhere convergence to for the Möbius-weighted averages along $\left( \lfloor n^c \rfloor \right)_{n \in \mathbb{N}}$ throughout $1<p<\infty$.