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Solving equations in dense Sidon sets

Part of: Sequences and sets

Published online by Cambridge University Press:  19 May 2021

SEAN PRENDIVILLE
SEAN PRENDIVILLE*
Affiliation:
Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF. e-mail: [email protected]
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Abstract

We offer an alternative proof of a result of Conlon, Fox, Sudakov and Zhao [CFSZ20] on solving translation-invariant linear equations in dense Sidon sets. Our proof generalises to equations in more than five variables and yields effective bounds.

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity
Secondary: 11B75: Other combinatorial number theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 1 , July 2022 , pp. 25 - 34
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

Bourgain, J.. On triples in arithmetic progression. Geom. Funct. Anal. 9(5) (1999), 968–984.CrossRefGoogle Scholar
Conlon, D., Fox, J., Sudakov, B. and Zhao, Y.. The regularity method for graphs with few 4-cycles. ArXiv e-prints (2020).CrossRefGoogle Scholar
Frankl, P., Graham, R. L. and Rödl, V.. Quantitative theorems for regular systems of equations. J. Combin. Theory Ser. A, 47(2) (1988), 246–261.CrossRefGoogle Scholar
Green, B.. Roth’s theorem in the primes. Ann. of Math. (2), 161(3) (2005), 1609–1636.CrossRefGoogle Scholar
Helfgott, H. A. and de Roton, A.. Improving Roth’s theorem in the primes. Int. Math. Res. Not. IMRN (4) (2011), 767–783.CrossRefGoogle Scholar
Naslund, E.. On improving Roth’s theorem in the primes. Mathematika 61(1) (2015), 49–62.CrossRefGoogle Scholar
O’Bryant, K.. A complete annotated bibliography of work related to Sidon sequences. Electron. J. Combin. (DS11) (2004).CrossRefGoogle Scholar
Schoen, T. and Sisask, O.. Roth’s theorem for four variables and additive structures in sums of sparse sets. Forum Math. Sigma 4 (2016) e5, 28.CrossRefGoogle Scholar
Tao, T. and Vu, V.. Additive combinatorics, volume 105 of Camb. Stud. Adv. Math. (Cambridge University Press, Cambridge, 2006).CrossRefGoogle Scholar
Vaughan, R. C.. The Hardy–Littlewood method, volume 125 of Cambridge Tracts in Math. (Cambridge University Press, Cambridge, second edition, 1997).CrossRefGoogle Scholar