On the maximal number of three-term arithmetic progressions in subsets of Z/pZ

TL;DR

This paper investigates the maximum density of 3-term arithmetic progressions in subsets of Z/pZ and proves that for small subset sizes, the maximum proportion follows a specific quadratic formula, with a structural characterization.

Contribution

It establishes that for small subset sizes, the maximum normalized count of 3-term progressions approaches a^2/2 and provides a structural theorem for such extremal sets.

Findings

Maximum proportion of 3-term progressions tends to a^2/2 for small a.

Structural description of sets with maximal 3-term progressions.

The limit c(a) equals a^2/2 for sufficiently small a.

Abstract

Let a be a real number between 0 and 1. Ernie Croot showed that the quantity \max_A #(3-term arithmetic progressions in A)/p^2, where A ranges over all subsets of Z/pZ of size at most a*p, tends to a limit as p tends to infinity through primes. Writing c(a) for this limit, we show that c(a) = a^2/2 provided that a is smaller than some absolute constant. In fact we prove rather more, establishing a structure theorem for sets having the maximal number of 3-term progressions amongst all subsets of Z/pZ of cardinality m, provided that m < c*p.