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

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A transference approach to a Roth-type theorem in the squares. / Prendiville, Sean; Browning, T.D.

In: International Mathematics Research Notices, Vol. 2017, No. 7, 2017, p. 2219–2248.

Research output: Contribution to journalJournal articlepeer-review

Harvard

Prendiville, S & Browning, TD 2017, 'A transference approach to a Roth-type theorem in the squares', International Mathematics Research Notices, vol. 2017, no. 7, pp. 2219–2248. https://doi.org/10.1093/imrn/rnw096

APA

Prendiville, S., & Browning, T. D. (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

Vancouver

Prendiville S, Browning TD. A transference approach to a Roth-type theorem in the squares. International Mathematics Research Notices. 2017;2017(7):2219–2248. https://doi.org/10.1093/imrn/rnw096

Author

Prendiville, Sean ; Browning, T.D. / A transference approach to a Roth-type theorem in the squares. In: International Mathematics Research Notices. 2017 ; Vol. 2017, No. 7. pp. 2219–2248.

Bibtex

@article{27fff3eb24a347629e1ce4446689b44e,
title = "A transference approach to a Roth-type theorem in the squares",
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.",
author = "Sean Prendiville and T.D. Browning",
year = "2017",
doi = "10.1093/imrn/rnw096",
language = "English",
volume = "2017",
pages = "2219–2248",
journal = "International Mathematics Research Notices",
issn = "1073-7928",
publisher = "Oxford University Press",
number = "7",

}

RIS

TY - JOUR

T1 - A transference approach to a Roth-type theorem in the squares

AU - Prendiville, Sean

AU - Browning, T.D.

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

U2 - 10.1093/imrn/rnw096

DO - 10.1093/imrn/rnw096

M3 - Journal article

VL - 2017

SP - 2219

EP - 2248

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 7

ER -