The structure of critical sets for F_p arithmetic progressions

TL;DR

This paper investigates functions on finite fields that minimize three-term arithmetic progression counts under density constraints, revealing their near indicator nature, local minimality, and relation to sumsets.

Contribution

It characterizes the structure of functions minimizing 3-term APs, showing they are nearly indicator functions linked to sumsets, advancing understanding of additive combinatorics in finite fields.

Findings

Functions are nearly indicator functions.

They satisfy a local minimality property.

They approximate indicator functions for sumsets A+B.

Abstract

Fix a prime p and a density 0 < d <= 1. Among all functions f : F_p -> [0,1], what can one say about those which assign minimal weight to three-term arithmetic progressions -- that is, the sum of f(a)f(a+x)f(a+2x) is minimal as we sum over all a and x -- subject to the density constraint that the expected value of f equals d? In the present paper we show three things about them: 1) Such f are nearly indicator functions; 2) They enjoy a certain ``local minimal'' property; and, 3) They are approximately indicator functions for certain sumsets A+B.