Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-19T19:26:17.864Z Has data issue: false hasContentIssue false
  • English
  • Français

A removal lemma for linear configurations in subsets of the circle

Part of: Sequences and sets

Published online by Cambridge University Press:  28 June 2013

Pablo Candela  and
Olof Sisask
Pablo Candela
Affiliation:
Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (pc308@cam.ac.uk)
Olof Sisask
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK (o.sisask@qmul.ac.uk)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We obtain a removal lemma for systems of linear equations over the circle group, using a similar result for finite fields due to Král′, Serra and Vena, and we discuss some applications.

Keywords

removal lemmatorussystems of linear equations

MSC classification

Secondary: 11B30: Arithmetic combinatorics; higher degree uniformity 11B75: Other combinatorial number theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 3 , October 2013 , pp. 657 - 666
Copyright
Copyright © Edinburgh Mathematical Society 2013 

References

1.Candela, P. and Sisask, O., On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime, Acta Math. Hungar. 132(3) (2011), 223–243.CrossRefGoogle Scholar
2.Deitmar, A. and Echterhoff, S., Principles of harmonic analysis (Springer, 2009).Google Scholar
3.Eisner, T. and Tao, T., Large values of the Gowers–Host–Kra seminorms, J. Analyse Math. 117 (2012), 133–186.CrossRefGoogle Scholar
4.Gowers, W. T. and Wolf, J., The true complexity of a system of linear equations, Proc. Lond. Math. Soc. 100 (2010), 155–176.CrossRefGoogle Scholar
5.Green, B. J., A Szemerédi-type regularity lemma in abelian groups, with applications, Geom. Funct. Analysis 15(2) (2005), 340–376.CrossRefGoogle Scholar
6.Král′, D., Serra, O. and Vena, L., On the removal lemma for linear systems over abelian groups, Eur. J Combin. 34 (2013), 248–259.CrossRefGoogle Scholar
7.Král′, D., Serra, O. and Vena, L., A removal lemma for systems of linear equations over finite fields, Israel J. Math. 187 (2012), 193–207.CrossRefGoogle Scholar
8.Lev, V. F., On sum-free subsets of the torus group, Funct. Approx. Comment. Math. 37(2) (2007), 277–283.CrossRefGoogle Scholar
9.Macbeath, A. M., On measure of sum sets, II, Proc. Camb. Phil. Soc. 49 (1953), 40–43.CrossRefGoogle Scholar
10.Raikov, D. A., On the addition of point-sets in the sense of Schnirelmann, Mat. Sb. 5(47) (1939), 425–440.Google Scholar
11.Ruzsa, I. Z., Sumsets and structure, in Combinatorial number theory and additive group theory, pp. 87–210, Advanced Courses in Mathematics—CRM Barcelona (Springer, 2009).Google Scholar
12.Shapira, A., A proof of Green's conjecture regarding the removal properties of sets of linear equations, J. Lond. Math. Soc. 81(2) (2010), 355–373.CrossRefGoogle Scholar
13.Tao, T. and Vu, V., Additive combinatorics (Cambridge University Press, 2006).CrossRefGoogle Scholar
14.Vaaler, J. D., A geometric inequality with applications to linear forms, Pac. J. Math. 83(2) (1979), 543–553.CrossRefGoogle Scholar