Two dimensional arithmetic progressions avoiding squares

TL;DR

This paper establishes an upper bound on the size of symmetric two-dimensional arithmetic progressions avoiding squares within a bounded interval, improving previous results and linking to quadratic non-residue bounds.

Contribution

It provides a new upper bound for symmetric 2D arithmetic progressions avoiding squares, advancing understanding in additive number theory.

Findings

Proper symmetric 2D progressions avoiding squares have at most O(T^{20/27+ε}) elements.

The result improves upon previous bounds by Croot, Lyall, and Rice.

Connections are discussed between these bounds and least quadratic non-residues.

Abstract

We show that any proper symmetric two dimensional arithmetic progression contained in the interval $[-T,T]$ which avoids non-zero perfect squares has at most $O_\varepsilon(T^{20/27+\varepsilon})$ elements. This improves on a result of Croot, Lyall and Rice. We also discuss lower bounds for this problem and their connections to bounds for the least quadratic non-residue modulo a prime.