Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T21:12:24.561Z Has data issue: false hasContentIssue false

Kneser’s theorem in $\sigma $-finite abelian groups

Published online by Cambridge University Press:  10 January 2022

Pierre-Yves Bienvenu
Affiliation:
Institute of Analysis and Number Theory, TU Graz, Kopernikusgasse 24/II, Graz 8010, Austria
François Hennecart*
Affiliation:
UJM-Saint-Étienne, CNRS, ICJ UMR 5208, University of Lyon, 23 rue du docteur Paul Michalon, Saint-Étienne 42023, France

Abstract

Let G be a $\sigma $ -finite abelian group, i.e., $G=\bigcup _{n\geq 1} G_n$ where $(G_n)_{n\geq 1}$ is a nondecreasing sequence of finite subgroups. For any $A\subset G$ , let $\underline {\mathrm {d}}( A ):=\liminf _{n\to \infty }\frac {|A\cap G_n|}{|G_n|}$ be its lower asymptotic density. We show that for any subsets A and B of G, whenever $\underline {\mathrm {d}}( A+B )<\underline {\mathrm {d}}( A )+\underline {\mathrm {d}}( B )$ , the sumset $A+B$ must be periodic, that is, a union of translates of a subgroup $H\leq G$ of finite index. This is exactly analogous to Kneser’s theorem regarding the density of infinite sets of integers. Further, we show similar statements for the upper asymptotic density in the case where $A=\pm B$ . An analagous statement had already been proven by Griesmer in the very general context of countable abelian groups, but the present paper provides a much simpler argument specifically tailored for the setting of $\sigma $ -finite abelian groups. This argument relies on an appeal to another theorem of Kneser, namely the one regarding finite sumsets in an abelian group.

Type
Article
Copyright
© Canadian Mathematical Society, 2022

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

References

Bihani, P. and Jin, R., Kneser’s theorem for upper Banach density. J. Théor. Nombres Bordeaux 18(2006), 323343.10.5802/jtnb.547CrossRefGoogle Scholar
Griesmer, J., Small-sum pairs for upper Banach density in countable abelian groups. Adv. Math. 246(2013), 220264.10.1016/j.aim.2013.06.005CrossRefGoogle Scholar
Halberstam, H. and Roth, K. F., Sequences. 2nd ed. Springer-Verlag, New York–Berlin, 1983, xviii+292 pp.10.1007/978-1-4613-8227-0CrossRefGoogle Scholar
Hamidoune, Y. and Rödseth, Ö., On bases for $\sigma$ -finite groups. Math. Scand. 78(1996), 246254.10.7146/math.scand.a-12586CrossRefGoogle Scholar
Hegyvári, N., On iterated difference sets in groups. Period. Math. Hung. 43(2001), 105110.10.1023/A:1015285615996CrossRefGoogle Scholar
Hegyvári, N. and Hennecart, F., Iterated difference sets in $\sigma$ -finite groups. Ann. Univ. Sci. Budapest, 50(2007), 17.Google Scholar
Jin, R., Solution to the inverse problem for upper asymptotic density. J. Reine Angew. Math. 595(2006), 121166.Google Scholar
Kneser, M., Abschätzungen der asymptotischen Dichte von Summenmengen. Math. Z. 58(1953), 459484.10.1007/BF01174162CrossRefGoogle Scholar