A Structure Theorem for Positive Density Sets Having the Minimal Number of 3-term Arithmetic Progressions

2003/05/22 by Ernie Croot, Croot, Ernie
Mathematics · #11P70 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Number Theory (math.NT) #math.CO #math.NT #msc:11P70

paper · pdf · doi:10.48550/arxiv.math/0305318

arxiv created 2003/05/22 · openalex publication_date 2003/05/22 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Assuming the well-known conjecture that [x,x+x^t] contains a prime for t > 0 and x sufficiently large, we prove: For 0 < r < 1, there exists 0 < s < r < 1, 0 < d < 1, and infinitely many primes q such that if S is a subset of Z/qZ having density at least s, and having the least number of 3-term arithemtic progressions among all sets of density at least s, then S is nearly translation invariant in a very strong sense. Namely, there exists 0 <= b <= q-1 such that |S intersect (S + bj)| = (1-g(s))|S|, for every 0 < j < q^d, where g(s) -> 0 as s -> 0. A curious feature of the proof is that Behrend's construction on large subsets of 1,2,...,x containing no 3-term a.p., is a key ingredient.

Citations

• The Difference Between Consecutive Primes, II
• On Certain Sets of Integers

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