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Sidon Sets are Proportionally Sidon with Small Sidon Constants

Part of: Harmonic analysis in one variable Abstract harmonic analysis

Published online by Cambridge University Press:  11 December 2018

Kathryn E. Hare  and
Robert (Xu) Yang
Kathryn E. Hare
Affiliation:
Dept. of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada Email: [email protected][email protected]
Robert (Xu) Yang
Affiliation:
Dept. of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada Email: [email protected][email protected]
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Abstract

In his seminal work on Sidon sets, Pisier found an important characterization of Sidonicity: A set is Sidon if and only if it is proportionally quasi-independent. Later, it was shown that Sidon sets were proportionally “special” Sidon in several other ways. Here, we prove that Sidon sets in torsion-free groups are proportionally $n$-degree independent, a higher order of independence than quasi-independence, and we use this to prove that Sidon sets are proportionally Sidon with Sidon constants arbitrarily close to one, the minimum possible value.

Keywords

Sidon setindependent set

MSC classification

Primary: 43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
Secondary: 42A55: Lacunary series of trigonometric and other functions; Riesz products
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 4 , December 2019 , pp. 798 - 809
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

This research was supported in part by NSERC grant 2016-03719. This paper is in final form and no version of it will be submitted for publication elsewhere.

References

Bourgain, J., Propriétés de décomposition pour les ensembles de Sidon . Bull. Soc. Math. France 111(1983), 421–428.Google Scholar
Bourgain, J., Subspaces of ℓ N ∞ , arithmetical diameter and Sidon sets . In: Probability in Banach spaces V , Lecture Notes in Math., 1153, Springer, Berlin, 1985, pp. 96–127. https://doi.org/10.1007/BFb0074947 Google Scholar
Bourgain, J., Sidon sets and Riesz products . Ann. Inst. Fourier (Grenoble) 35(1985), 137–148.Google Scholar
Graham, C. C. and Hare, K. E., Characterizing Sidon sets by interpolation properties of subsets . Colloq. Math. 112(2008), 175–199. https://doi.org/10.4064/cm112-2-1 Google Scholar
Graham, C. C. and Hare, K. E., Interpolation and Sidon sets for compact groups , CMS Books in Math., Springer, New York, 2013. https://doi.org/10.1007/978-1-4614-5392-5 Google Scholar
Grow, D., Sidon sets and I 0 sets . Colloq. Math. 53(1987), 269–270. https://doi.org/10.4064/cm-53-2-269-270 Google Scholar
Hare, K. E. and Ramsey, L. T., The relationship between 𝜀-Kronecker and Sidon sets . Canad. Math. Bull. 59(2016), 521–527. https://doi.org/10.4153/CMB-2016-002-3 Google Scholar
Li, D. and Queffélec, H., Introduction à l’etude des espaces de Banach. Analyse et probabilités , Cours Spécalisés, 12, Société Mathématique de France, Paris, 2004.Google Scholar
López, J. and Ross, K., Sidon sets , Lecture Notes in Pure and Applied Mathematics, 13, Marcel Dekker, New York, 1975.Google Scholar
Malliavin-Brameret, M. P. and Malliavin, P., Caractérisation arithmétique des ensembles de Helson . C.R.Acad. Sci. Paris Sér. A–B 264(1967), A192–A193.Google Scholar
Neuwirth, S., The maximum modulus of a trigonometric trinomial . J. Anal. Math. 104(2008), 371–396. https://doi.org/10.1007/s11854-008-0028-2 Google Scholar
Pisier, G., De nouvelles caractérisations des ensembles de Sidon . In: Mathematical analysis and applications, Part B , Adv. Math. Suppl. Studies, 7b, Academic Press, New York-London, 1981, pp. 685–726.Google Scholar
Pisier, G., Conditions d’ entropie et caractérisations arithmétiques des ensembles de Sidon . Proc. Conf. on Modern Topics in Harmonic Analysis (Torino/Milano) . Ist. Naz. Alta Mat. Francesco Severi, Rome, 1983, pp. 911–944.Google Scholar
Pisier, G., Arithmetic characterizations of Sidon sets . Bull. Amer. Math. Soc. 8(1983), 87–89. https://doi.org/10.1090/S0273-0979-1983-15092-9 Google Scholar
Ramsey, L. T., Comparisons of Sidon and I 0 sets . Colloq. Math. 70(1996), 103–132. https://doi.org/10.4064/cm-70-1-103-132 Google Scholar