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On the density of Kravitz sets

Vsevolod Lev, Máté Matolcsi, Péter Pál Pach, Dániel Varga

TL;DR

This paper determines density thresholds for subsets $A$ of the finite field $\mathbb{F}_p$ such that $A+A-2A$ equals the whole group. It proves that for primes $p>3$, any $A$ with $|A|>\frac{2}{7}p$ satisfies $A+A-2A=\mathbb{F}_p$, and it provides two independent proofs: a linear programming bound using a witness set $X=\{0,2,3,4,7,8,9,10\}$ (with modulus conditions) and a purely combinatorial local-constraint argument. The results refute a related conjecture on sums of dilates for the case $(1,1,-2)$, and the paper discusses avenues to push the bound toward $\frac{1}{4}p$ (or higher) via Fourier-analytic refinements and alternative witness constructions. The work also situates the problem within the broader framework of the geometric fractional chromatic number and the study of $A$-sumsets in finite fields, with detailed computational verification and an extensive appendix detailing the linear-programming machinery.

Abstract

We show that for a subset $A$ of the cyclic group of prime order $p>3$, if the sumset $A+A-2A$ is not the whole group, then $|A|\le \frac27\,p$. Besides combinatorial arguments, we utilize a general technique involving linear programming.

On the density of Kravitz sets

TL;DR

This paper determines density thresholds for subsets of the finite field such that equals the whole group. It proves that for primes , any with satisfies , and it provides two independent proofs: a linear programming bound using a witness set (with modulus conditions) and a purely combinatorial local-constraint argument. The results refute a related conjecture on sums of dilates for the case , and the paper discusses avenues to push the bound toward (or higher) via Fourier-analytic refinements and alternative witness constructions. The work also situates the problem within the broader framework of the geometric fractional chromatic number and the study of -sumsets in finite fields, with detailed computational verification and an extensive appendix detailing the linear-programming machinery.

Abstract

We show that for a subset of the cyclic group of prime order , if the sumset is not the whole group, then . Besides combinatorial arguments, we utilize a general technique involving linear programming.
Paper Structure (6 sections, 1 theorem, 23 equations, 1 figure)

This paper contains 6 sections, 1 theorem, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Linear programming proof of Theorem \ref{['t:main']}
  3. Combinatorial proof of Theorem \ref{['t:main']}
  4. Concluding remarks
  5. Appendix
  6. Acknowledgments

Key Result

Theorem 1

If $p>3$ is a prime and $A\subseteq \mathbb{F}_p$ is a subset of size $|A|>\frac{2}{7}\,p$, then $A+A-2A=\mathbb{F}_p$.

Figures (1)

  • Figure 1: A 23 element witness set for $p=151$ and $d=2$, giving a density upper bound $5/19$. Removing any of the elements results in a weaker bound.

Theorems & Definitions (3)

  • Conjecture 1: Pon13
  • Theorem 1
  • Remark