Mathematica Pannonica
Volume/Issue:
Volume 29_NS3: Issue 2
Kahane’s Upper Density and Syndetic Sets in LCA Groups
Author:
Szilárd Gy. Révész HUN-REN Alfréd Rényi Institute of Mathematics, H-1053 Budapest, Hungary

Search for other papers by Szilárd Gy. Révész in
Current site
Google Scholar
PubMed Close
Pages:
268–281
Online Publication Date:
05 Dec 2023
Publication Date:
07 Dec 2023
Article Category:
Research Article
DOI:
https://doi.org/10.1556/314.2023.00028
Keywords:
Density; asymptotic uniform upper density; locally compact Abelian group; difference set; syndetic set
Open access

Asymptotic uniform upper density, shortened as a.u.u.d., or simply upper density, is a classical notion which was first introduced by Kahane for sequences in the real line.

Syndetic sets were defined by Gottschalk and Hendlund. For a locally compact group 𝐺, a set 𝑆 ⊂ 𝐺 is syndetic, if there exists a compact subset 𝐶 ⋐ 𝐺 such that 𝑆𝐶 = 𝐺. Syndetic sets play an important role in various fields of applications of topological groups and semigroups, ergodic theory and number theory. A lemma in the book of Fürstenberg says that once a subset 𝐴 ⊂ ℤ has positive a.u.u.d., then its difference set 𝐴 − 𝐴 is syndetic.

The construction of a reasonable notion of a.u.u.d. in general locally compact Abelian groups (LCA groups for short) was not known for long, but in the late 2000’s several constructions were worked out to generalize it from the base cases of ℤ𝑑 and ℝ𝑑. With the notion available, several classical results of the Euclidean setting became accessible even in general LCA groups.

Here we work out various versions in a general locally compact Abelian group 𝐺 of the classical statement that if a set 𝑆 ⊂ 𝐺 has positive asymptotic uniform upper density, then the difference set 𝑆 − 𝑆 is syndetic.

  • [1]

    Beurling, A. Interpolation for an interval in R1. In: The Collected Works of Arne Beurling, in: Harmonic Analysis, vol. 2. Birkhauser Boston, Boston, MA, 1989.

    • Search Google Scholar
    • Export Citation
  • [2]

    Beurling, A. Balayage of Fourier-Stieltjes transforms. in: The Collected Works of Arne Beurling, in: Harmonic Analysis, vol. 2, Birkhauser Boston, Boston, MA, 1989.

    • Search Google Scholar
    • Export Citation
  • [3]

    Berdysheva, E. and Révész, Sz. Gy. Delsarte’s extremal problem and packing on locally compact Abelian groups. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XXIV (2023), 1007–1052.

    • Search Google Scholar
    • Export Citation
  • [4]

    Cohn, H., Kumar, A., Miller, S. D., Radchenko, D., and Viazovska, M. The sphere packing problem in dimension 24. Ann. of Math. (3)85, 3 (2017), 1017–1033.

    • Search Google Scholar
    • Export Citation
  • [5]

    Fürstenberg, H. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, 1981.

  • [6]

    Erdős, P. and Stone, A. H. On the sum of two Borel sets. Proc. Amer. Math. Soc. 25 (1970), 304–306.

  • [7]

    Gottschalk, W. H. and Hedlund, G. A. Topological dynamics. American Mathematical Society Colloquium Publications 36, American Mathematical Society, Providence, R. I., 1955. vii+151 pp.

    • Search Google Scholar
    • Export Citation
  • [8]

    Gröchenig, K., Kutyniok, G., and Seip, K. Landau’s necessary density conditions for LCA groups. J. Funct. Anal. 255 (2008), 1831–1850.

    • Search Google Scholar
    • Export Citation
  • [9]

    Groemer, H. Existenzsätze für Lagerungen im Euklidischen Raum. (German.) Math. Z. 81 (1963), 260–278.

  • [10]

    Hegyvári, N. On iterated difference sets in groups. Period. Math. Hungar. 43 1-2, (2001), 105–110.

  • [11]

    Hewitt, E. and Ross, K. A. Abstract harmonic analysis, I. Die Grundlehren der mathematischen Wissenchaften, Band 115. Springer Verlag, Berlin, Göttingen, Heidelberg, 1963.

    • Search Google Scholar
    • Export Citation
  • [12]

    Hewitt, E. and Ross, K. A. Abstract harmonic analysis, II. Die Grundlehren der mathematischen Wissenchaften, Band 152. Springer Verlag, Berlin, Heidelberg, New York, Budapest, 1970.

    • Search Google Scholar
    • Export Citation
  • [13]

    Kahane, J.-P. Sur les fonctions moyenne-périodiques bornées. Ann. Inst. Fourier 7 (1957), 293–314.

  • [14]

    Kahane, J.-P. Sur quelques problèmes d’unicité et de prolongement, relatifs aux fonctions approchables par des sommes d’exponentielles. Thése de doctorat, Université de Paris, Série A, no. 2633, 1954, 102 pages.

    • Search Google Scholar
    • Export Citation
  • [15]

    Kahane, J.-P. Sur quelques problèmes d’unicité et de prolongement, relatifs aux fonctions approchables par des sommes d’exponentielles. Ann. Inst. Fourier 5 (1954), 39–130.

    • Search Google Scholar
    • Export Citation
  • [16]

    Kolountzakis, M. N., Sz. Gy. Révész, Turán’s extremal problem for positive definite functions on groups. J. London Math. Soc. 74 (2006), 475–496.

    • Search Google Scholar
    • Export Citation
  • [17]

    Landau, H. J. Necessary density conditions for sampling and interpolation of certain entire functions. Acta Math. 117 (1967), 37–52.

  • [18]

    Révész, Sz. Gy. On asymptotic uniform upper density in locally compact Abelian groups, preprint. See on arXive as arXiv 0904.1567, (2009).

    • Search Google Scholar
    • Export Citation
  • [19]

    Révész, Sz. Gy. Turán’s extremal problem on locally compact Abelian groups. Anal. Math. 37 (2011), 15–50.

  • [20]

    Révész, Sz. Gy. Extremal problems for positive definite functions and polynomials. Thesis for the “Doctor of the Academy” degree, April 2009. See at http://www.renyi.hu/~revesz/preprints.html.

    • Search Google Scholar
    • Export Citation
  • [21]

    Révész, Sz. Gy. On asymptotic uniform upper density in locally compact Abelian groups. Real Analysis Exchange 37, 1 (2012), 24–31. (In the Supplement 35th Summer Symposium Conference Reports.)

    • Search Google Scholar
    • Export Citation
  • [22]

    Rogers, C. A. A linear Borel set whose difference set is not a Borel set. Bull. London Math. Soc. 2 (1970), 41–42.

  • [23]

    Rudin, W. Fourier analysis on groups. Interscience Tracts in Pure and Applied Mathematics, No. 12. Interscience Publishers (a division of John Wiley and Sons), New York-London 1962 ix+285 pp.

    • Search Google Scholar
    • Export Citation
  • [24]

    Ruzsa, I. Z. On difference-sequences. Acta Arith. XXV (1974), 151–157.

  • [25]

    Totik, V. On comparision of asymptotic uniform upper densities in non-discrete LCA groups. e-mail to Sz. Révész, 25 July 2010; also in the Referee report on the thesis “Extremal problems for positive definite functions and polynomials” by Sz. Gy. Révész, Hungarian academy of Sciences, Budapest, 2011.

    • Search Google Scholar
    • Export Citation
  • [26]

    Viazovska, M. S. The sphere packing problem in dimension 8. Ann. of Math. 185, 3 (2017), 991–1015.

  • Collapse
  • Expand
Cover Mathematica Pannonica
Mathematica Pannonica
Print ISSN:
0865-2090
Online ISSN:
2786-0752

Editor in Chief: László TÓTH, University of Pécs, Pécs, Hungary

Honorary Editors in Chief:

  • † István GYŐRI, University of Pannonia, Veszprém, Hungary
  • János PINTZ, HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • Ferenc SCHIPP, Eötvös Loránd University, Budapest, Hungary and University of Pécs, Pécs, Hungary
  • Sándor SZABÓ, University of Pécs, Pécs, Hungary
     

Deputy Editors in Chief:

  • Erhard AICHINGER, JKU Linz, Linz, Austria
  • Ferenc HARTUNG, University of Pannonia, Veszprém, Hungary
  • Ferenc WEISZ, Eötvös Loránd University, Budapest, Hungary

Editorial Board

  • Attila BÉRCZES, University of Debrecen, Debrecen, Hungary
  • István BERKES, Rényi Institute of Mathematics, Budapest, Hungary
  • Károly BEZDEK, University of Calgary, Calgary, Canada
  • György DÓSA, University of Pannonia, Veszprém, Hungary
  • Balázs KIRÁLY – Managing Editor, University of Pécs, Pécs, Hungary
  • Vedran KRCADINAC, University of Zagreb, Zagreb, Croatia 
  • Željka MILIN ŠIPUŠ, University of Zagreb, Zagreb, Croatia
  • Gábor NYUL, University of Debrecen, Debrecen, Hungary
  • Margit PAP, University of Pécs, Pécs, Hungary
  • István PINK, University of Debrecen, Debrecen, Hungary
  • Mihály PITUK, University of Pannonia, Veszprém, Hungary
  • Lukas SPIEGELHOFER, Montanuniversität Leoben, Leoben, Austria
  • Andrea ŠVOB, University of Rijeka, Rijeka, Croatia
  • Csaba SZÁNTÓ, Babeş-Bolyai University, Cluj-Napoca, Romania
  • Jörg THUSWALDNER, Montanuniversität Leoben, Leoben, Austria
  • Zsolt TUZA, University of Pannonia, Veszprém, Hungary

Advisory Board

  • Szilárd RÉVÉSZ – Chair, HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • Gabriella BÖHM
  • György GÁT, University of Debrecen, Debrecen, Hungary

University of Pécs,
Faculty of Sciences,
Institute of Mathematics and Informatics
Department of Mathematics
7624 Pécs, Ifjúság útja 6., HUNGARY
(36) 72-503-600 / 4179
ltoth@gamma.ttk.pte.hu

  • Mathematical Reviews
  • Zentralblatt
  • DOAJ

Publication Model Gold Open Access
Online only
Submission Fee none
Article Processing Charge 0 EUR/article (temporarily)
Subscription Information Gold Open Access

Mathematica Pannonica
Language English
Size A4
Year of
Foundation		 1990
Volumes
per Year		 1
Issues
per Year		 2
Founder Akadémiai Kiadó
Founder's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Publisher Akadémiai Kiadó
Publisher's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible
Publisher		 Chief Executive Officer, Akadémiai Kiadó
ISSN 2786-0752 (Online)
ISSN 0865-2090 (Print)

Monthly Content Usage

Abstract Views Full Text Views PDF Downloads
Aug 2024 0 89 10
Sep 2024 0 64 4
Oct 2024 0 115 8
Nov 2024 0 156 8
Dec 2024 0 39 9
Jan 2025 0 44 1
Feb 2025 0 0 0