Clusters of primes with square-free translates

Abstract

Let $\mathcal{R}$ be a finite set of integers satisfying appropriate local conditions. We show the existence of long clusters of primes $p$ in bounded length intervals with $p-b$ squarefree for all $b\in \mathcal{R}$. Moreover, we can enforce that the primes $p$ in our cluster satisfy any one of the following conditions: (1) $p$ lies in a short interval $[N,N+N^{\frac{7}{12}+\hat{I}\mu }]$, (2) $p$ belongs to a given inhomogeneous Beatty sequence, (3) with $c\in (\frac{8}{9},1)$ fixed, $p^c$ lies in a prescribed interval mod $1$ of length $p^{-1+c+\hat{I}\mu }$.

15 pages