Polynomial configurations and pointwise averages along Piatetski-Shapiro sequences

Abstract

In this paper, we prove that for every integer $k\geq2$ and every $c>1$ sufficiently close to $1$, there is $\kappa>0$ such that every sufficiently large subset of $\{1,\ldots,N\}$ of density at least $(\log\log N)^{-\kappa}$ contains \[ x,\quad x+\lfloor n^c\rfloor,\quad x+\lfloor n^c\rfloor^2, \quad\ldots,\quad x+\lfloor n^c\rfloor^k. \] We also prove pointwise almost-everywhere convergence of the associated multiple ergodic averages.