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

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


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.