Final published version
Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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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 -