Polynomial configurations and pointwise averages along Piatetski--Shapiro primes

For every \(c>1\) sufficiently close to \(1\), we prove that there is \(κ>0\) such that every sufficiently large \(A⊂[N]\) of density at least \((log N)-κ\) contains \[ x,\ x+(p-1),\ x+(p-1)^2,…,\ x+(p-1)^k, p= n^c∈ P. \] We also prove pointwise almost-everywhere convergence of the corresponding multiple ergodic averages.