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Rado's criterion over squares and higher powers

Research output: Contribution to Journal/Magazine › Journal article › peer-review

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Rado's criterion over squares and higher powers. / Chow, Sam; Lindqvist, Sofia; Prendiville, Sean.
In: Journal of the European Mathematical Society, Vol. 23, No. 6, 16.02.2021, p. 1925-1997.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Chow, S, Lindqvist, S & Prendiville, S 2021, 'Rado's criterion over squares and higher powers', Journal of the European Mathematical Society, vol. 23, no. 6, pp. 1925-1997. https://doi.org/10.4171/JEMS/1047

APA

Chow, S., Lindqvist, S., & Prendiville, S. (2021). Rado's criterion over squares and higher powers. Journal of the European Mathematical Society, 23(6), 1925-1997. https://doi.org/10.4171/JEMS/1047

Vancouver

Chow S, Lindqvist S, Prendiville S. Rado's criterion over squares and higher powers. Journal of the European Mathematical Society. 2021 Feb 16;23(6):1925-1997. doi: 10.4171/JEMS/1047

Author

Chow, Sam ; Lindqvist, Sofia ; Prendiville, Sean. / Rado's criterion over squares and higher powers. In: Journal of the European Mathematical Society. 2021 ; Vol. 23, No. 6. pp. 1925-1997.

Bibtex

@article{80370ed602e9467e924a1a1ee92d3cf4,
title = "Rado's criterion over squares and higher powers",
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.",
keywords = ". Arithmetic combinatorics, arithmetic Ramsey theory, Weyl sums, smooth numbers, restriction theory",
author = "Sam Chow and Sofia Lindqvist and Sean Prendiville",
year = "2021",
month = feb,
day = "16",
doi = "10.4171/JEMS/1047",
language = "English",
volume = "23",
pages = "1925--1997",
journal = "Journal of the European Mathematical Society",
issn = "1435-9855",
publisher = "European Mathematical Society Publishing House",
number = "6",

}

RIS

TY - JOUR

T1 - Rado's criterion over squares and higher powers

AU - Chow, Sam

AU - Lindqvist, Sofia

AU - Prendiville, Sean

PY - 2021/2/16

Y1 - 2021/2/16

N2 - 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.

AB - 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.

KW - . Arithmetic combinatorics

KW - arithmetic Ramsey theory

KW - Weyl sums

KW - smooth numbers

KW - restriction theory

U2 - 10.4171/JEMS/1047

DO - 10.4171/JEMS/1047

M3 - Journal article

VL - 23

SP - 1925

EP - 1997

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 6

ER -