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Vinogradov's theorem for primes with restricted digits

James Leng, Mehtaab Sawhney

TL;DR

This work extends Vinogradov’s three-primes theorem to primes with digits restricted in a large alphabet base $g$. It combines a refined Fourier-analytic framework for digit-restricted sets with zero-density–based approximants for primes, and introduces digit-carry conditioning into product-measure decompositions to enable Maynard-type analysis. The authors isolate a main-term that factors into local arithmetic components and count representations within the restricted-digit set, showing correction terms from zeros contribute negligibly on average. The result generalizes prior digit-restriction results and demonstrates a robust method for handling primes with digital constraints in a Vinogradov-type setting.

Abstract

Let $g$ be sufficiently large, $b\in\{0,\ldots,g-1\}$, and $\mathcal{S}_b$ be the set of integers with no digit equal to $b$ in their base $g$ expansion. We prove that every sufficiently large odd integer $N$ can be written as $p_1 + p_2 + p_3$ where $p_i$ are prime and $p_i\in \mathcal{S}_b$.

Vinogradov's theorem for primes with restricted digits

TL;DR

This work extends Vinogradov’s three-primes theorem to primes with digits restricted in a large alphabet base . It combines a refined Fourier-analytic framework for digit-restricted sets with zero-density–based approximants for primes, and introduces digit-carry conditioning into product-measure decompositions to enable Maynard-type analysis. The authors isolate a main-term that factors into local arithmetic components and count representations within the restricted-digit set, showing correction terms from zeros contribute negligibly on average. The result generalizes prior digit-restriction results and demonstrates a robust method for handling primes with digital constraints in a Vinogradov-type setting.

Abstract

Let be sufficiently large, , and be the set of integers with no digit equal to in their base expansion. We prove that every sufficiently large odd integer can be written as where are prime and .
Paper Structure (13 sections, 30 theorems, 195 equations)

This paper contains 13 sections, 30 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Sketch of proof
  3. Notation
  4. Acknowledgements
  5. Combinatorics of digits representations
  6. Fourier analytic estimates for restricted digit sets
  7. Divisor functions estimates for sets with missing digits
  8. Fourier approximant of the primes and zero-density estimates
  9. Proof of \ref{['thm:main']} given \ref{['lem:correction', 'lem:main-term']}
  10. Proof of Lemmas 5.3 and 5.4
  11. Handling correction terms
  12. Computing main term
  13. Character sum estimates

Key Result

Theorem 1.1

Fix $A\ge 1$, $g$ be sufficiently large and $b\in \{0,\ldots,g-1\}$. Let $\mathbf{P}$ denote the set of primes and $\mathcal{S}_b$ denote the set of positive integers with no $b$ in their base-$g$ expansion. Then

Theorems & Definitions (62)

  • Theorem 1.1
  • Remark
  • Lemma 2.1
  • Remark
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Remark
  • proof
  • ...and 52 more