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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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.

Item Type:
Journal Article
Journal or Publication Title:
International Mathematics Research Notices
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600
Subjects:
ID Code:
137145
Deposited By:
Deposited On:
01 Oct 2019 12:55
Refereed?:
Yes
Published?:
Published
Last Modified:
24 Sep 2020 04:54