Partition regularity and multiplicatively syndetic sets

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

Abstract

We show how multiplicatively syndetic sets can be used in the study of partition regularity of dilation invariant systems of polynomial equations. In particular, we prove that a dilation invariant system of polynomial equations is partition regular if and only if it has a solution inside every multiplicatively syndetic set. We also adapt the methods of Green–Tao and Chow–Lindqvist–Prendiville to develop a syndetic version of Roth’s density increment strategy. This argument is then used to obtain bounds on the Rado numbers of configurations of the form $\{x,d,x+d,x+2d\}$.
Original languageEnglish
Pages (from-to)109-138
JournalActa Arithmetica
Volume196
DOIs
Publication statusPublished - 19 Jun 2020

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