S\'ark\"ozy's theorem for shifted primes with restricted digits

Abstract

For a base $b\geq 2$ and a set of digits $\mathcal{A}\subset \{0,...,b-1\}$, let $\mathcal{P}$ denote the set of prime numbers with digits restricted to $\mathcal{A}$, when written in base-$b$. We prove that if $A\subset \mathbb{N}$ has positive upper Banach density, then there exists a prime $p\in \mathcal{P}$ and two elements $a_1,a_2\in A$ such that $a_2=a_1+p-1$. The key ingredients are the Furstenberg correspondence principle and a discretized Hardy-Littlewood circle method used by Maynard. As a byproduct of our work, we prove a Dirichlet-type theorem for the distribution of $\mathcal{P}$ in residue classes, and a Vinogradov-type theorem for the decay of associated exponential sums. These estimates arise from the unique structure of associated Fourier transforms, which take the form of Riesz products.