A transference approach to a Roth-type theorem in the squares

TL;DR

This paper demonstrates that subsets of square numbers with positive density contain solutions to certain linear equations, establishing new bounds and partition regularity results for equations in five or more variables using transference methods.

Contribution

It introduces a transference approach to prove a Roth-type theorem for squares in five or more variables, improving previous bounds that required seven or more variables.

Findings

Subsets of squares with positive density contain solutions to translation-invariant linear equations in five or more variables.

Established the partition regularity of diagonal quadrics with coefficients summing to zero in five or more variables.

Used Green's transference technology to transfer bounds from linear to quadratic settings.

Abstract

We show that any subset of the squares of positive relative upper density contains non-trivial solutions to a translation-invariant linear equation in five or more variables, with explicit quantitative bounds. As a consequence, we establish the partition regularity of any diagonal quadric in five or more variables whose coefficients sum to zero. Unlike previous approaches, which are limited to equations in seven or more variables, we employ transference technology of Green to import bounds from the linear setting.