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

T.D. Browning, S. Prendiville, International Mathematics Research Notices 2017 (2017) 2219–2248.


Journal Article | Published
Author
Browning, Timothy DIST Austria; Prendiville, Sean M
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.
Publishing Year
Date Published
2017-04-01
Journal Title
International Mathematics Research Notices
Acknowledgement
Whilst working on this paper the authors were supported by ERC grant 306457.
Volume
2017
Issue
7
Page
2219 - 2248
IST-REx-ID
268

Cite this

Browning TD, Prendiville S. A transference approach to a Roth-type theorem in the squares. International Mathematics Research Notices. 2017;2017(7):2219-2248. doi:10.1093/imrn/rnw096
Browning, T. D., & Prendiville, S. (2017). A transference approach to a Roth-type theorem in the squares. International Mathematics Research Notices, 2017(7), 2219–2248. https://doi.org/10.1093/imrn/rnw096
Browning, Timothy D, and Sean Prendiville. “A Transference Approach to a Roth-Type Theorem in the Squares.” International Mathematics Research Notices 2017, no. 7 (2017): 2219–48. https://doi.org/10.1093/imrn/rnw096.
T. D. Browning and S. Prendiville, “A transference approach to a Roth-type theorem in the squares,” International Mathematics Research Notices, vol. 2017, no. 7, pp. 2219–2248, 2017.
Browning TD, Prendiville S. 2017. A transference approach to a Roth-type theorem in the squares. International Mathematics Research Notices. 2017(7), 2219–2248.
Browning, Timothy D., and Sean Prendiville. “A Transference Approach to a Roth-Type Theorem in the Squares.” International Mathematics Research Notices, vol. 2017, no. 7, Oxford University Press, 2017, pp. 2219–48, doi:10.1093/imrn/rnw096.
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