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

Research output: Contribution to journalJournal articlepeer-review

Published
<mark>Journal publication date</mark>2017
<mark>Journal</mark>International Mathematics Research Notices
Issue number7
Volume2017
Number of pages30
Pages (from-to)2219–2248
Publication StatusPublished
<mark>Original language</mark>English

Abstract

We show that any subset of the squares of positive relative upper density contains nontrivial 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.