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An inverse theorem on sets with rich additive structure modulo primes

Ernie Croot, Junzhe Mao, Chi Hoi Yip

TL;DR

The paper addresses an inverse sieve problem: given S ⊆ [N] whose reductions mod many primes lie in small arithmetic progressions, what global structure must S possess? The authors prove a sharp inverse-sieve theorem asserting that under suitable density and bound assumptions on the mod-p residue sets, the set A of integers up to N resonant with all R_p is an arithmetic progression of controlled length; they also develop refinements for when R_p are intervals or when unions of several short progressions are allowed. Beyond the main theorem, the work extends to longer progressions, unions of progressions, and inverse results for generalized arithmetic progressions (GAPs), accompanied by constructions showing sharpness and several applications to improved larger sieve bounds. These results deepen the connection between local additive-structure mod primes and global additive-structure, with potential algorithmic implications and broader applicability to sieve methods in analytic number theory.

Abstract

In this paper, we prove several results on the structure of maximal sets $S \subseteq [N]$ such that $S$ mod $p$ is contained in a short arithmetic progression, or the union of short progressions, where $p$ ranges over a subset of primes in an interval $[y,2y]$ with $(\log N)^{O(1)} < y \leq N$. We also provide several constructions demonstrating the sharpness of our results. Furthermore, as an application, we provide several improvements on the larger sieve bound for $|S|$ when $S$ mod $p$ has strong additive structure, parallel to the work of Green--Harper and Shao for improvements on the large sieve.

An inverse theorem on sets with rich additive structure modulo primes

TL;DR

The paper addresses an inverse sieve problem: given S ⊆ [N] whose reductions mod many primes lie in small arithmetic progressions, what global structure must S possess? The authors prove a sharp inverse-sieve theorem asserting that under suitable density and bound assumptions on the mod-p residue sets, the set A of integers up to N resonant with all R_p is an arithmetic progression of controlled length; they also develop refinements for when R_p are intervals or when unions of several short progressions are allowed. Beyond the main theorem, the work extends to longer progressions, unions of progressions, and inverse results for generalized arithmetic progressions (GAPs), accompanied by constructions showing sharpness and several applications to improved larger sieve bounds. These results deepen the connection between local additive-structure mod primes and global additive-structure, with potential algorithmic implications and broader applicability to sieve methods in analytic number theory.

Abstract

In this paper, we prove several results on the structure of maximal sets such that mod is contained in a short arithmetic progression, or the union of short progressions, where ranges over a subset of primes in an interval with . We also provide several constructions demonstrating the sharpness of our results. Furthermore, as an application, we provide several improvements on the larger sieve bound for when mod has strong additive structure, parallel to the work of Green--Harper and Shao for improvements on the large sieve.

Paper Structure

This paper contains 24 sections, 23 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Other extensions
  4. Future Directions and Questions
  5. Some odds and ends
  6. Preliminaries
  7. Six remarks and results concerning Theorem \ref{['maintheorem']}
  8. The length $(2y)^{1/2-\varepsilon} + 1$ on $A$ is best-possible
  9. The lower bound $y \gg (\log N)^{1/2\varepsilon}$ is a necessary condition
  10. The upper-bound on $|R_p|$ of size $p^{1/2-\varepsilon}$ is essentially necessary
  11. Theorem \ref{['maintheorem']} is false if we restrict to certain thinner subsets of the primes
  12. A construction showing that $S$ can be an arithmetic progression stretching nearly to $N$
  13. Theorem \ref{['maintheorem']} does not extend to polynomial progressions
  14. Proof of Theorems \ref{['maintheorem']} and \ref{['thm:interval']}
  15. Longer progressions: Proof of Theorems \ref{['thm:AP1/2']} and \ref{['thm:largeA']}
  16. ...and 9 more sections

Key Result

Theorem 1.1

There is an absolute constant $C>0$ such that the following holds. Let $\varepsilon \in (0, 1/2)$. Suppose $N$ is a positive integer with $N> N_0(\varepsilon)$, and $y$ satisfies Let ${\mathcal{P}}$ be a subset of primes in $[y,2y]$ satisfying For each prime $p \in {\mathcal{P}}$, choose a prescribed arithmetic progression $R_p \subseteq {\mathbb Z}_p$ with $|R_p| \leq p^{1/2-\varepsilon}+1$. Le

Theorems & Definitions (40)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 30 more