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  • 1806.05002

    Accepted author manuscript, 772 KB, PDF document

    Available under license: CC BY-NC: Creative Commons Attribution-NonCommercial 4.0 International License

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Rado's criterion over squares and higher powers

Research output: Contribution to journalJournal articlepeer-review

Forthcoming
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<mark>Journal publication date</mark>24/09/2019
<mark>Journal</mark>Journal of the European Mathematical Society
Volume0
Pages (from-to)0-0
Publication StatusAccepted/In press
<mark>Original language</mark>English

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 kth powers, provided the equation has at least (1+o(1))klogk 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.