Polynomial configurations and pointwise averages along Piatetski--Shapiro primes

Abstract

For every \(c>1\) sufficiently close to \(1\), we prove that there is \(\kappa>0\) such that every sufficiently large \(A\subset[N]\) of density at least \((\log N)^{-\kappa}\) contains \[ x,\ x+(p-1),\ x+(p-1)^2,\ldots,\ x+(p-1)^k, \qquad p=\lfloor n^c\rfloor\in\mathbb P. \] We also prove pointwise almost-everywhere convergence of the corresponding multiple ergodic averages.