Approximate invariance for ergodic actions of amenable groups. (English) Zbl 1432.37003

Björklund, Michael; Fish, Alexander

Discrete Anal. 2019, Paper No. 6, 56 p. (2019).

Summary: We develop in this paper some general techniques to analyze action sets of small doubling for probability measure-preserving actions of amenable groups.
As an application of these techniques, we prove a dynamical generalization of Kneser’s celebrated density theorem for subsets in \((\mathbb{Z},+)\), valid for any countable amenable group, and we show how it can be used to establish a plethora of new inverse product set theorems for upper and lower asymptotic densities. We provide several examples demonstrating that our results are optimal for the settings under study.

MSC:

37A05	Dynamical aspects of measure-preserving transformations
37A25	Ergodicity, mixing, rates of mixing
37A30	Ergodic theorems, spectral theory, Markov operators
22D40	Ergodic theory on groups

Keywords:

action sets; aperiodicity; density theorems

References:

[1]	Vitaly Bergelson and Hillel Furstenberg. WM groups and Ramsey theory. Topology Appl., 156(16):2572-2580, 2009. 40 · Zbl 1231.05270
[2]	Michael Björklund. Small product sets in compact groups. Fund. Math., 238(1):1-27, 2017. 13, 25, 26, 46, 50
[3]	Michael Björklund. Product set phenomena for measured groups. Ergodic Theory Dynam. Systems, 38(8):2913-2941, 2018. 47, 48 · Zbl 1401.37005
[4]	Michael Björklund and Alexander Fish. Product set phenomena for countable groups. Adv. Math., 275:47-113, 2015. 50 · Zbl 1395.37008
[5]	Michael Björklund and Alexander Fish. Ergodic theorems for coset spaces. J. Anal. Math., 135(1):85-122, 2018. 49 · Zbl 1402.37007
[6]	Michael Björklund and Tobias Hartnick. Analytic properties of approximate lattices. Arxiv preprint arXiv:1709.09942. 3 · Zbl 1496.22004
[7]	Michael Björklund and Tobias Hartnick. Approximate lattices. Duke Math. J., 167(15):2903-2964, 2018. 3 · Zbl 1452.22002
[8]	Michael Björklund, Tobias Hartnick, and Felix Pogorzelski. Aperiodic order and spherical diffraction, I: auto-correlation of regular model sets. Proc. Lond. Math. Soc. (3), 116(4):957-996, 2018. 3 · Zbl 1394.52022
[9]	Jean Bourgain and Alex Gamburd. Uniform expansion bounds for Cayley graphs of SL 2 (F p ). Ann. of Math. (2), 167(2):625-642, 2008. 3 · Zbl 1216.20042
[10]	Emmanuel Breuillard, Ben Green, and Terence Tao. The structure of approximate groups. Publ. Math. Inst. Hautes Études Sci., 116:115-221, 2012. 3 · Zbl 1260.20062
[11]	A. J. Chintschin. Drei Perlen der Zahlentheorie. Akademie-Verlag, Berlin, 1984. Reprint of the 1951 translation from the Russian, With a foreword by Helmut Koch. 3
[12]	Manfred Einsiedler and Thomas Ward. Homogeneous dynamics: a study guide. In Introduction to modern mathematics, volume 33 of Adv. Lect. Math. (ALM), pages 171-201. Int. Press, Somerville, MA, 2015. 14, 38, 47 · Zbl 1364.37007
[13]	G. A. Freȋman. Foundations of a structural theory of set addition. American Mathematical Society, Providence, R. I., 1973. Translated from the Russian, Translations of Mathematical Monographs, Vol 37. 3 · Zbl 0271.10044
[14]	Harry Furstenberg. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Anal. Math., 31:204-256, 1977. 48 · Zbl 0347.28016
[15]	Eli Glasner. Ergodic theory via joinings, volume 101 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2003. 15, 17, 32 · Zbl 1038.37002
[16]	John T. Griesmer. Small-sum pairs for upper Banach density in countable abelian groups. Adv. Math., 246:220-264, 2013. 9 · Zbl 1297.11131
[17]	H. A. Helfgott. Growth and generation in SL 2 (Z/pZ). Ann. of Math. (2), 167(2):601-623, 2008. 3 · Zbl 1213.20045
[18]	Karl H. Hofmann and Sidney A. Morris. The structure of compact groups, volume 25 of De Gruyter Studies in Mathematics. De Gruyter, Berlin, 2013. A primer for the student-a handbook for the expert, Third edition, revised and augmented. 50 · Zbl 1277.22001
[19]	Renling Jin. Characterizing the structure of A + B when A + B has small upper Banach density. J. Number Theory, 130(8):1785-1800, 2010. 9 · Zbl 1201.11016
[20]	J. H. B. Kemperman. On products of sets in a locally compact group. Fund. Math., 56:51-68, 1964. 3, 25, 26
[21]	Martin Kneser. Abschätzung der asymptotischen Dichte von Summenmengen. Math. Z., 58:459-484, 1953. 3, 9 · Zbl 0051.28104
[22]	Martin Kneser. Summenmengen in lokalkompakten abelschen Gruppen. Math. Z., 66:88-110, 1956. 3, 25, 26
[23]	George W. Mackey. Ergodic transformation groups with a pure point spectrum. Illinois J. Math., 8:593-600, 1964. 15 · Zbl 0255.22014
[24]	Henry B. Mann. A proof of the fundamental theorem on the density of sums of sets of positive integers. Ann. of Math. (2), 43:523-527, 1942. 3, 9 · Zbl 0061.07406
[25]	Yves Meyer. Algebraic numbers and harmonic analysis. North-Holland Publishing Co., Amsterdam-London; · Zbl 0267.43001
[26]	American Elsevier Publishing Co., Inc., New York, 1972. North-Holland Mathematical Library, Vol. 2. 3
[27]	Alan L. T. Paterson. Amenability, volume 29 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1988. 46
[28]	Terence Tao. Product set estimates for non-commutative groups. Combinatorica, 28(5):547-594, 2008. 3 · Zbl 1254.11017
[29]	André Weil. L’intégration dans les groupes topologiques et ses applications. Actual. Sci. Ind., no. 869. Hermann et Cie., Paris, 1940. [This book has been republished by the author at Princeton, N. J., 1941.]. 26 · JFM 66.1205.02
[30]	Reinhard Winkler. Hartman sets, functions and sequences-a survey. In Probability and number theory-Kanazawa 2005, volume 49 of Adv. Stud. Pure Math., pages 517-543. Math. Soc. Japan, Tokyo, 2007. 6, 47
[31]	Robert J. Zimmer. Ergodic theory and semisimple groups, volume 81 of Monographs in Mathematics. Birkhäuser Verlag, Basel, 1984 · Zbl 0571.58015