×

Nonstandard analysis and the sumset phenomenon in arbitrary amenable groups. (English) Zbl 1320.03092

Summary: M. Beiglböck et al. [Adv. Math. 223, No. 2, 416–432 (2010; Zbl 1187.43002)] proved that if subsets \(A\), \(B\) of a countable discrete amenable group \(G\) have positive Banach densities \(\alpha\) and \(\beta\) respectively, then the product set \(AB\) is piecewise syndetic, that is, there exists \(k\) such that the union of \(k\)-many left translates of \(AB\) is thick. Using nonstandard analysis, we give a shorter alternative proof of this result that does not require \(G\) to be countable and moreover yields the explicit bound \(k\leq 1/\alpha\beta\). We also prove with similar methods that if \(\{A_{i}\}_{i=1}^{n}\) are finitely many subsets of \(G\) having positive Banach densities \(\alpha_{i}\) and \(G\) is countable, then there exists a subset \(B\) whose Banach density is at least \(\prod_{i=1}^{n}\alpha_{i}\) and such that \(BB^{-1}\subseteq\bigcap_{i=1}^{n}A_{i}A_{i}^{-1}\). In particular, the latter set is piecewise Bohr.

MSC:

03H05 Nonstandard models in mathematics
43A07 Means on groups, semigroups, etc.; amenable groups
11B05 Density, gaps, topology
11B13 Additive bases, including sumsets

Citations:

Zbl 1187.43002
PDFBibTeX XMLCite
Full Text: arXiv Euclid

References:

[1] \beginbbook \beditor\binitsL. O. \bsnmArkeryd, \beditor\binitsN. J. \bsnmCutland and \beditor\binitsC. W. \bsnmHenson, eds., \bbtitleNonstandard analysis: Theory and applications, \bsertitleNATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. \bseriesno493, \bpublisherKluwer Academic Publishers Group, \blocationDordrecht, \byear1997. \endbbook \OrigBibText Leif O. Arkeryd, Nigel J. Cutland, and C. Ward Henson (eds.), Nonstandard analysis , NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 493, Dordrecht, Kluwer Academic Publishers Group, 1997, Theory and applications. \endOrigBibText \bptokstructpyb \endbibitem Mathematical Reviews (MathSciNet):
[2] \beginbarticle \bauthor\binitsM. , \bauthor\binitsV. \bsnmBergelson and \bauthor\binitsA. \bsnmFish, \batitleSumset phenomenon in countable amenable groups, \bjtitleAdv. Math. \bvolume223 (\byear2010), no. \bissue2, page 416-\blpage432. \endbarticle \OrigBibText Mathias Beiglböck, Vitaly Bergelson, and Alexander Fish, Sumset phenomenon in countable amenable groups , Adv. Math. 223 (2010), no. 2, 416-432. \endOrigBibText \bptokstructpyb \endbibitem Mathematical Reviews (MathSciNet): · Zbl 1187.43002
[3] \beginbchapter \bauthor\binitsV. \bsnmBergelson, \bauthor\binitsN. \bsnmHindman and \bauthor\binitsR. \bsnmMcCutcheon, \bctitleNotions of size and combinatorial properties of quotient sets in semigroups, \bbtitleProceedings of the 1998 topology and dynamics conference (Fairfax, VA), Topology Proc., vol. \bseriesno23, \byear1998, pp. page 23-\blpage60. \endbchapter \OrigBibText Vitaly Bergelson, Neil Hindman, and Randall McCutcheon, Notions of size and combinatorial properties of quotient sets in semigroups , Proceedings of the 1998 Topology and Dynamics Conference (Fairfax, VA), vol. 23, 1998, pp. 23-60. \endOrigBibText \bptokstructpyb \endbibitem Mathematical Reviews (MathSciNet):
[4] \beginbbook \bauthor\binitsC. C. \bsnmChang and \bauthor\binitsH. J. \bsnmKeisler, \bbtitleModel theory, \bedition3rd ed., \bsertitleStudies in Logic and the Foundations of Mathematics, vol. \bseriesno73, \bpublisherNorth-Holland Publishing Co., \blocationAmsterdam, \byear1990. \endbbook \OrigBibText Chen Chung Chang and H. Jerome Keisler, Model theory , third ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland Publishing Co., Amsterdam, 1990. \endOrigBibText \bptokstructpyb \endbibitem Mathematical Reviews (MathSciNet):
[5] \beginbbook \bauthor\binitsN. J. \bsnmCutland, \bbtitleLoeb measures in practice: Recent advances, \bsertitleLecture Notes in Mathematics, vol. \bseriesno1751, \bpublisherSpringer, \blocationBerlin, \byear2000. \endbbook \OrigBibText Nigel J. Cutland, Loeb measures in practice: recent advances , Lecture Notes in Mathematics, vol. 1751, Springer-Verlag, Berlin, 2000. \endOrigBibText \bptokstructpyb \endbibitem Mathematical Reviews (MathSciNet): · Zbl 0963.28015
[6] \beginbbook \beditor\binitsN. J. \bsnmCutland, \beditor\binitsM. \bparticleDi \bsnmNasso and \beditor\binitsD. A. \bsnmRoss, eds., \bbtitleNonstandard methods and applications in mathematics, \bsertitleLecture Notes in Logic, vol. \bseriesno25, \bpublisherAssociation for Symbolic Logic, \blocationLa Jolla, CA, \byear2006. \endbbook \OrigBibText Nigel J. Cutland, Mauro Di Nasso, and David A. Ross (eds.), Nonstandard methods and applications in mathematics , Lecture Notes in Logic, vol. 25, Association for Symbolic Logic, La Jolla, CA, 2006. \endOrigBibText \bptokstructpyb \endbibitem Mathematical Reviews (MathSciNet):
[7] \beginbarticle \bauthor\binitsM. \bsnmDi Nasso, \batitleEmbeddability properties of difference sets, \bjtitleIntegers \bvolume14 (\byear2014), no. \bissueA27, page 1-\blpage24. \endbarticle \OrigBibText Mauro Di Nasso, Embeddability properties of difference sets , Integers, 14 (2014) A27, 1-24 \endOrigBibText \bptokstructpyb \endbibitem
[8] \beginbarticle \bauthor\binitsE. , \batitleOn groups with full Banach mean value, \bjtitleMath. Scand. \bvolume3 (\byear1955), page 243-\blpage254. \endbarticle \OrigBibText Erling Følner, On groups with full Banach mean value , Math. Scand. 3 (1955), 243-254. \endOrigBibText \bptokstructpyb \endbibitem Mathematical Reviews (MathSciNet):
[9] \beginbbook \bauthor\binitsR. \bsnmGoldblatt, \bbtitleLectures on the hyperreals: An introduction to nonstandard analysis, \bsertitleGraduate Texts in Mathematics, vol. \bseriesno188, \bpublisherSpringer, \blocationNew York, \byear1998. \endbbook \OrigBibText Robert Goldblatt, Lectures on the hyperreals , Graduate Texts in Mathematics, vol. 188, Springer-Verlag, New York, 1998, An introduction to nonstandard analysis. \endOrigBibText \bptokstructpyb \endbibitem Mathematical Reviews (MathSciNet):
[10] \beginbarticle \bauthor\binitsC. W. \bsnmHenson, \batitleOn the nonstandard representation of measures, \bjtitleTrans. Amer. Math. Soc. \bvolume172 (\byear1972), page 437-\blpage446. \endbarticle \OrigBibText C. Ward Henson, On the nonstandard representation of measures , Trans. Amer. Math. Soc. 172 (1972), 437-446. \endOrigBibText \bptokstructpyb \endbibitem
[11] \beginbarticle \bauthor\binitsN. \bsnmHindman and \bauthor\binitsD. \bsnmStrauss, \batitleDensity in arbitrary semigroups, \bjtitleSemigroup Forum \bvolume73 (\byear2006), no. \bissue2, page 273-\blpage300. \endbarticle \OrigBibText Neil Hindman and Dona Strauss, Density in arbitrary semigroups , Semigroup Forum 73 (2006), no. 2, 273-300. \endOrigBibText \bptokstructpyb \endbibitem Mathematical Reviews (MathSciNet): · Zbl 1111.22004
[12] \beginbarticle \bauthor\binitsR. \bsnmJin, \batitleThe sumset phenomenon, \bjtitleProc. Amer. Math. Soc. \bvolume130 (\byear2002), no. \bissue3, page 855-\blpage861 \bcomment(electronic). \endbarticle \OrigBibText Renling Jin, The sumset phenomenon , Proc. Amer. Math. Soc. 130 (2002), no. 3, 855-861 (electronic). \endOrigBibText \bptokstructpyb \endbibitem Mathematical Reviews (MathSciNet): · Zbl 0985.03066
[13] \beginbarticle \bauthor\binitsE. \bsnmLindenstrauss, \batitlePointwise theorems for amenable groups, \bjtitleInvent. Math. \bvolume146 (\byear2001), no. \bissue2, page 259-\blpage295. \endbarticle \OrigBibText Elon Lindenstrauss, Pointwise theorems for amenable groups , Invent. Math. 146 (2001), no. 2, 259-295. \endOrigBibText \bptokstructpyb \endbibitem Mathematical Reviews (MathSciNet): · Zbl 1038.37004
[14] \beginbbook \bauthor\binitsA. L. T. \bsnmPaterson, \bbtitleAmenability, \bsertitleMathematical Surveys and Monographs, vol. \bseriesno29, \bpublisherAmer. Math. Soc., \blocationProvidence, RI, \byear1988. \endbbook \OrigBibText Alan L. T. Paterson, Amenability , Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. \endOrigBibText \bptokstructpyb \endbibitem Mathematical Reviews (MathSciNet):
[15] \beginbbook \bauthor\binitsS. \bsnmWagon, \bbtitleThe Banach-Tarski paradox, \bpublisherCambridge University Press, \blocationCambridge, \byear1993. \bcommentWith a foreword by Jan Mycielski. Corrected reprint of the 1985 original. \endbbook \OrigBibText Stan Wagon, The Banach-Tarski paradox , Cambridge University Press, Cambridge, 1993, With a foreword by Jan Mycielski, Corrected reprint of the 1985 original. \endOrigBibText \bptokstructpyb \endbibitem Mathematical Reviews (MathSciNet):
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.