Polynomial configurations and pointwise averages along Piatetski-Shapiro sequences

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