Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-17T16:10:17.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Triangles in Cartesian Squares of Quasirandom Groups

Part of: Extremal combinatorics Measure-theoretic ergodic theory

Published online by Cambridge University Press:  25 August 2016

V. BERGELSON ,
D. ROBERTSON  and
P. ZORIN-KRANICH
V. BERGELSON
Affiliation:
Department of Mathematics, Ohio State University, 231 W. 18th Avenue, Columbus OH 43212, USA (e-mail: [email protected], [email protected])
D. ROBERTSON
Affiliation:
Department of Mathematics, Ohio State University, 231 W. 18th Avenue, Columbus OH 43212, USA (e-mail: [email protected], [email protected])
P. ZORIN-KRANICH
Affiliation:
Universität Bonn, Mathematisches Institut, Endenicher Allee 60, D-53115 Bonn, Germany (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that triangular configurations are plentiful in large subsets of Cartesian squares of finite quasirandom groups from classes having the quasirandom ultraproduct property, for example the class of finite simple groups. This is deduced from a strong double recurrence theorem for two commuting measure-preserving actions of a minimally almost periodic (not necessarily amenable or locally compact) group on a (not necessarily separable) probability space.

MSC classification

Primary: 05D10: Ramsey theory
Secondary: 28D15: General groups of measure-preserving transformations
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 2 , March 2017 , pp. 161 - 182
Copyright
Copyright © Cambridge University Press 2016 

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.)

References

[1] Austin, T. (2010) On the norm convergence of non-conventional ergodic averages. Ergodic Theory Dynam. Systems 30 321–338.Google Scholar
[2] Austin, T. (2013) Non-conventional ergodic averages for several commuting actions of an amenable group. J. Analyse Math., to appear.Google Scholar
[3] Austin, T. (2015) Quantitative equidistribution for certain quadruples in quasi-random groups. Combin. Probab. Comput. 24 376–381. With erratum.Google Scholar
[4] Austin, T. (2016) Ajtai–Szemerédi theorems over quasirandom groups. In Recent Trends in Combinatorics, Vol. 159 (Beveridge, A., Griggs, R. J., Hogben, L., Musiker, G. and Tetali, P., eds), Springer International Publishing, pp. 453–484.Google Scholar
[5] Babai, L. and Sós, V. T. (1985) Sidon sets in groups and induced subgraphs of Cayley graphs. European J. Combin. 6 101–114.Google Scholar
[6] Bergelson, V. (2000) Ergodic theory and Diophantine problems. In Topics in Symbolic Dynamics and Applications: Temuco 1997, Vol. 279 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 167–205.Google Scholar
[7] Bergelson, V. (2003) Minimal idempotents and ergodic Ramsey theory. In Topics in Dynamics and Ergodic theory, Vol. 310 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 8–39.Google Scholar
[8] Bergelson, V. and Furstenberg, H. (2009) WM groups and Ramsey theory. Topology Appl. 156 2572–2580.Google Scholar
[9] Bergelson, V. and Hindman, N. (1990) Nonmetrizable topological dynamics and Ramsey theory. Trans. Amer. Math. Soc. 320 293–320.CrossRefGoogle Scholar
[10] Bergelson, V. and McCutcheon, R. (2007) Central sets and a non-commutative Roth theorem. Amer. J. Math. 129 1251–1275.Google Scholar
[11] Bergelson, V. and Tao, T. (2014) Multiple recurrence in quasirandom groups. Geom. Funct. Anal. 24 1–48.CrossRefGoogle Scholar
[12] Bergelson, V., Christopherson, C., Robertson, D. and Zorin-Kranich, P. (2016) Finite products sets and minimally almost periodic groups. J. Funct. Anal. 270 2126–2167.Google Scholar
[13] Bergelson, V., McCutcheon, R. and Zhang, Q. (1997) A Roth theorem for amenable groups. Amer. J. Math. 119 1173–1211.CrossRefGoogle Scholar
[14] Chu, Q. (2011) Multiple recurrence for two commuting transformations. Ergodic Theory Dynam. Systems 31 771–792.Google Scholar
[15] Chu, Q. and Zorin-Kranich, P. (2015) Lower bound in the Roth theorem for amenable groups. Ergodic Theory Dynam. Systems 35 1746–1766.Google Scholar
[16] Cutland, N. J. (1983) Nonstandard measure theory and its applications. Bull. London Math. Soc. 15 529–589.Google Scholar
[17] de Leeuw, K. and Glicksberg, I. (1961) Applications of almost periodic compactifications. Acta Math. 105 63–97.Google Scholar
[18] Ellis, R. (1958), Distal transformation groups. Pacific J. Math. 8 401–405.Google Scholar
[19] Furstenberg, H. (1977) Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Analyse Math. 31 204–256.Google Scholar
[20] Furstenberg, H. (1981), Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures, Princeton University Press.Google Scholar
[21] Gowers, W. T. (2008) Quasirandom groups. Combin. Probab. Comput. 17 363–387.Google Scholar
[22] Phelps, R. R. (2001) Lectures on Choquet's Theorem, second edition, Vol. 1757 of Lecture Notes in Mathematics, Springer.Google Scholar
[23] Robinson, D. J. S. (1996) A Course in the Theory of Groups, second edition, Vol. 80 of Graduate Texts in Mathematics, Springer.Google Scholar
[24] Schnell, C. (2007) Idempotent ultrafilters and polynomial recurrence. arXiv:0711.0484 Google Scholar
[25] Tserunyan, A. (2014) Mixing and triple recurrence in probability groups. arXiv:1405.5629 Google Scholar
[26] Yang, Y. (2016) Ultraproducts of quasirandom groups with small cosocles. J. Group Theory, to appear.Google Scholar