The Minimal Number of Three-Term Arithmetic Progressions Modulo a Prime Converges to a Limit

2004/12/31 by Ernie Croot, Croot, Ernie
Mathematics · #11P99 #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematics and Applications #Number Theory (math.NT) #math.CO #math.NT #msc:11P99

paper · pdf · doi:10.48550/arxiv.math/0501004

This draft is a significantly shorter proof of the theorem. To appear in Canadian Math Bulletin

openalex publication_date 2004/12/31 · arxiv created 2006/07/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a density t in (0,1], and a prime p, let S be any subset of F_p having at least tp elements, and having the least number of three-term arithmetic progressions mod p among all subsets of F_p with at least tp elements. Define N(t,p) to be 1/p² times the number of three-term arithmetic progressions in S modulo p. Note that N(t,p) does not depend on S -- it only depends on t and p. An old result of Varnavides shows that for fixed t, N(t,p) > c(t) > 0 for all primes p sufficiently large. But, does N(t,p) converge to a limit as p -> infinity? We prove that it does.

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