Ergodic averages for sparse sequences along primes

We investigate the limiting behavior of multiple ergodic averages along sparse sequences evaluated at prime numbers. Our sequences arise from smooth and well-behaved functions that have polynomial growth. Central to this topic is a comparison result between standard Cesàro averages along positive integers and averages weighted by the (modified) von Mangoldt function. The main ingredients are a recent result of Matomäki, Shao, Tao and Teräväinen on the Gowers uniformity of the latter function in short intervals, a lifting argument that allows one to pass from actions of integers to flows, a simultaneous (variable) polynomial approximation in appropriate short intervals, and some quantitative equidistribution results for the former polynomials. We derive numerous applications in multiple recurrence, additive combinatorics, and equidistribution in nilmanifolds along primes. In particular, we deduce that any set of positive density contains arithmetic progressions with step $ \lfloor p^c \rfloor $, where $ c $ is a positive non-integer and $ p $ denotes a prime, establishing a conjecture of Frantzikinakis.