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

Prendiville, Sean and Browning, T.D. (2017) A transference approach to a Roth-type theorem in the squares. International Mathematics Research Notices, 2017 (7). 2219–2248. ISSN 1073-7928

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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.

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Journal Article
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International Mathematics Research Notices
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01 Oct 2019 12:55
Last Modified:
22 Nov 2022 08:07