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Partition regularity and multiplicatively syndetic sets

Volume 196 / 2020

Jonathan Chapman Acta Arithmetica 196 (2020), 109-138 MSC: Primary 11B30; Secondary 05D10. DOI: 10.4064/aa190421-11-3 Published online: 19 June 2020

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 $\{x,d,x+d,x+2d\}$.

Authors

  • Jonathan ChapmanDepartment of Mathematics
    University of Manchester
    Oxford Road
    Manchester, M13 9PL, UK
    e-mail

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