Sárközy's theorem for shifted primes with restricted digits

TL;DR

This work advances Sárközy-type recurrence questions for primes with digits restricted to a base-$b$ Cantor-like set by proving that any density-positive set $A\subset\mathbb{N}$ contains two elements separated by a restricted prime gap $p-1$, i.e. $a_2=a_1+p-1$ with $p\in\mathbb{P}_{\mathcal{C}}$, under explicit digit-structure conditions on $\mathcal{C}$ (including $1\in\mathcal{A}$ and a decomposition of excluded digits into intervals with $b-s>(k+1)b^{4/5+\varepsilon}$). The novel combination of the Furstenberg correspondence principle and a discretized Hardy–Littlewood circle method (in the Maynard tradition) yields both a Dirichlet-type equidistribution result for $\mathbb{P}_{\mathcal{C}}$ in residue classes and a Vinogradov-type decay bound for exponential sums over primes in $\mathcal{C}$, with the Fourier transforms presenting as Riesz products. Byproduct results include a Dirichlet-type theorem for $\mathbb{P}_{\mathcal{C}}$ in arithmetic progressions and a vanishing-exponential-sum bound for irrational frequencies, together with the observation that $\mathbb{P}_{\mathcal{C}}-1$ forms a van der Corput set. These developments connect ergodic-theoretic methods with explicit exponential-sum estimates in a digit-restricted setting, contributing to both structure theory and quantitative Sárközy-type phenomena in sparse, restricted-digit primes.

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.

Theorems & Definitions (39)

• Theorem 2: Maynard, maynard2019primesrestricteddigits
• Theorem 3
• Theorem 4
• Corollary 5
• Theorem 6
• Proposition 7
• Theorem 8
• proof : Proof of Proposition \ref{['prop:furstenberg']} assuming Theorem \ref{['thm:uniformDistribution']}
• Lemma 10
• proof