JournalsjemsVol. 23, No. 6pp. 1925–1997

Rado's criterion over squares and higher powers

  • Sam Chow

    University of Oxford, UK
  • Sofia Lindqvist

    University of Oxford, UK
  • Sean Prendiville

    University of Manchester, UK
Rado's criterion over squares and higher powers cover

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Abstract

We establish partition regularity of the generalised Pythagorean equation in five or more variables. Furthermore, we show how Rado’s characterisation of a partition regular equation remains valid over the set of positive kkth powers provided the equation has at least (1+o(1))k(1+o(1))k log kk variables. We thus completely describe which diagonal forms are partition regular and which are not, given sufficiently many variables. In addition, we prove a supersaturated version of Rado’s theorem for a linear equation restricted either to squares minus one or to logarithmically-smooth numbers.

Cite this article

Sam Chow, Sofia Lindqvist, Sean Prendiville, Rado's criterion over squares and higher powers. J. Eur. Math. Soc. 23 (2021), no. 6, pp. 1925–1997

DOI 10.4171/JEMS/1047