Skip to main content Accessibility help
×
Home
Hostname: page-component-78dcdb465f-2ktwh Total loading time: 0.244 Render date: 2021-04-19T07:18:11.813Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Sets of zero discrete harmonic density

Published online by Cambridge University Press:  20 November 2009

COLIN C. GRAHAM
Affiliation:
Department of Mathematics, University of British Columbia, V6T 1Y4 Vancouver, B.C., Canada. e-mail: ccgraham@alum.mit.edu
KATHRYN E. HARE
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1 Canada. e-mail: kehare@uwaterloo.ca
Corresponding

Abstract

Let G be a compact, connected, abelian group with dual group Γ. The set E has zero discrete harmonic density (z.d.h.d.) if for every open UG and μ ∈ Md(G) there exists ν ∈ Md(U) with = on E. I0 sets in the duals of these groups have z.d.h.d. We give properties of such sets, exhibit non-Sidon sets having z.d.h.d., and prove union theorems. In particular, we prove that unions of I0 sets have z.d.h.d. and provide a new approach to two long-standing problems involving Sidon sets.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2009

Access options

Get access to the full version of this content by using one of the access options below.

References

[1]Bourgain, J. Subspaces of l N, arithmetical diameter and Sidon sets. in Probability in Banach spaces, V (Medford, Mass., 1984), 96127. Lecture Notes in Math. 1153 (Springer, Berlin, 1985).Google Scholar
[2]Déchamps–Gondim, M.Ensembles de Sidon topologiques. Ann. Inst. Fourier (Grenoble) 22, fasc.3 (1972).CrossRefGoogle Scholar
[3]Déchamps, M. and Selles, O.Compacts associés aux sommes de suites lacunaires. Publ. Math. Orsay 01 (1996) 2740Google Scholar
[4]Gaposhkin, V. F.A uniqueness theorem for multiple lacunary trigonometric series. Mat. Zametki 16 (1974) 865870 (Translated in Math. Notes 16 (1974) 1112–1115)Google Scholar
[5]Graham, C. C. and Hare, K. E.ϵ-Kronecker and I 0 sets in abelian groups, I: arithmetic properties of ϵ-Kronecker sets. Math. Proc. Camb. Phil. Soc. 140 (2006), no. 3, 475489.CrossRefGoogle Scholar
[6]Graham, C. C. and Hare, K. E.ϵ-Kronecker and I 0 sets in abelian groups, III: Interpolation by measures on small sets, Studia Math. 171 (2005), no. 1, 1532.CrossRefGoogle Scholar
[7]Graham, C. C., Hare, K. E. and Ramsey, L. T.Union problems for I 0 sets. Acta Sci. Math. (Szeged) 75 (2009), 175195.Google Scholar
[8]Graham, C. C. and McGehee, O. CarruthEssays in Commutative Harmonic Analysis. (Springer-Verlag 1979).CrossRefGoogle Scholar
[9]Hadamard, J.Essai sur l'étude des fonctions données par leur développement de Taylor. J. Math. Pures Appl. (4) 8 (1892) 101186Google Scholar
[10]Kahane, J.-P. and Katznelson, Y.Entiers aléatoires et analyse harmonique. J. Anal. Math. 105 (2008), 363378.CrossRefGoogle Scholar
[11]Kalton, J. N.On Vector-valued inequalities for Sidon sets and sets of interpolation. Colloq. Math. 54 (1993) 233244CrossRefGoogle Scholar
[12]Lopez, J. and Ross, K.Sidon sets. Lecture Notes in Pure and Applied Math. 13 (Marcel Dekker 1975).Google Scholar
[13]Méla, J.-F.Approximation diophantine et ensembles lacunaires. Mem. Bull. Math. Soc. France 19 (1969) 2654Google Scholar
[14]Ramsey, L. T.A theorem of C. Ryll-Nardzewski and metrizable l.c.a. groups. Proc. Amer. Math. Soc. 78 (1980) no. 2, 221224CrossRefGoogle Scholar
[15]Rudin, W.Fourier Analysis on Groups. (Wiley Interscience 1962).Google Scholar
[16]Ryll-Nardzewski, C.Concerning almost periodic extensions of functions. Colloq. Math. 12 (1964) 235237CrossRefGoogle Scholar
[17]Shapiro, G. S.Balayage in Fourier transforms: general results, perturbation, and balayage with sparse frequencies. Trans. Amer. Math. Soc. 225 (1977), 183198.CrossRefGoogle Scholar
[18]Shapiro, G. S.Unique balayage in Fourier transforms on compact abelian groups. Proc. Amer. Math. Soc. 70 (1978), no. 2, 146150.CrossRefGoogle Scholar
[19]Zygmund, A.On a theorem of Hadamard. Ann. Soc. Polon. Math. 21 (1948) 5269Google Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 17 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 19th April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Sets of zero discrete harmonic density
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Sets of zero discrete harmonic density
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Sets of zero discrete harmonic density
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *