Rado's criterion over squares and higher powers

Chow, Sam and Lindqvist, Sofia and Prendiville, Sean (2021) Rado's criterion over squares and higher powers. Journal of the European Mathematical Society, 23 (6). pp. 1925-1997. ISSN 1435-9855

Text (1806.05002)
1806.05002.pdf - Accepted Version
Available under License Creative Commons Attribution.
Download (790kB)

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.

Item Type:

Journal Article

Journal or Publication Title:

Journal of the European Mathematical Society

Uncontrolled Keywords:

/dk/atira/pure/subjectarea/asjc/2600/2604

Subjects:

?? . arithmetic combinatoricsarithmetic ramsey theoryweyl sumssmooth numbersrestriction theoryapplied mathematicsgeneral mathematicsmathematics(all) ??

Departments:

Faculty of Science and Technology > Mathematics and Statistics

ID Code:

137152

Deposited By:

ep_importer_pure

Deposited On:

27 Sep 2019 14:15

Refereed?:

Yes

Published?:

Published

Last Modified:

23 Jan 2025 02:35

URI:

https://eprints.lancs.ac.uk/id/eprint/137152