2025/10/15 by Alex B. Burgin, Burgin, Alex
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2510.13076
openalex publication_date 2025/10/15 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28
For a base b≥ 2 and a set of digits A⊂ \0,...,b-1\, let P denote the set of prime numbers with digits restricted to A, when written in base-b. We prove that if A⊂ ℕ has positive upper Banach density, then there exists a prime p∈ P and two elements a1,a2∈ A such that a2=a1+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 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.