A Transference Approach to a Roth-Type Theorem in the Squares

Tim D Browning, Sean M Prendiville

Research output: Contribution to journalArticle (Academic Journal)peer-review

8 Citations (Scopus)
189 Downloads (Pure)


We show that any subset of the squares of positive relative upper density contains non-trivial 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.
Original languageEnglish
Pages (from-to)2219-2248
Number of pages30
JournalInternational Mathematics Research Notices
Issue number7
Early online date13 Jun 2016
Publication statusPublished - Apr 2017

Bibliographical note

25 pages


  • math.NT
  • math.CO
  • 11B30
  • 11D09
  • 11P55


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