Primes with Restricted-Digit Differences

Abstract

Fix a base $b\ge 2$ and a digit set $\mathcal{D}\subset \{0,1,\dots ,b-1\}$, and let ${\mathcal{C}}_k$ be the set of integers in $[0,b^k)$ whose $k$ base-$b$ digits all lie in $\mathcal{D}$. We study prime pairs and three-term arithmetic progressions in primes whose difference, or common difference, belongs to ${\mathcal{C}}_k$. Under a natural digit non-resonance condition, we establish a localized Fourier criterion guaranteeing asymptotic formulae with explicit leading constants for both problems. Using a transfer-operator argument, we verify this criterion in explicit parameter ranges for several natural families of digit sets.