Character-free approach to progression-free sets

TL;DR

This paper introduces a simple combinatorial method to estimate the maximum density of sets without arithmetic progressions in finite vector spaces, showing it scales inversely with the dimension.

Contribution

It provides an elementary combinatorial proof that the density of progression-free sets in finite vector spaces is bounded by a constant times 1/r, offering a new perspective.

Findings

Progression-free set density is O(1/r) in finite vector spaces.

Elementary combinatorial argument suffices for the bound.

Simplifies previous complex proofs.

Abstract

We present an elementary combinatorial argument showing that the density of a progression-free set in a finite r-dimensional vector space is O(1/r).