Pointwise Ergodic Theorems Along pᶜ for Every $c>1$

Let $c>1$ be noninteger, and let $Λ$ denote the von Mangoldt function. The averages \[ A_Nᶜ f(x):=1N∑ₙ₌₁N Λ(n)f(Tn^cx) \] satisfy the variational estimate \[ \|V^r(A_Nᶜ f:N)\|L^p(X) c,p,r\|f\|L^p(X) \] for every $12$, 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 ( nᶜ )n ∈ N throughout $1