Abstract
We study the relationships between three different classes of sequences (or sets) of integers, namely rigidity sequences, Kazhdan sequences (or sets) and nullpotent sequences. We prove that rigidity sequences are non-Kazhdan and nullpotent, and that all other implications are false. In particular, we show by probabilistic means that there exist sequences of integers which are both nullpotent and Kazhdan. Moreover, using Baire category methods, we provide general criteria for a sequence of integers to be a rigidity sequence. Finally, we give a new proof of the existence of rigidity sequences which are dense in ℤ for the Bohr topology, a result originally due to Griesmer.
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J. Aaronson, M. Hosseini and M. Lemańczyk, IP-rigidity and eigenvalue groups, Ergodic Theory Dynam. Systems 34 (2014), 1057–1076.
T. Adams, Tower multiplexing and slow weak mixing, Colloq. Math. 138 (2015), 47–72.
C. Badea and S. Grivaux, Kazhdan sets in groups and equidistribution properties, J. Funct. Anal. 273 (2017), 1931–1969.
C. Badea and S. Grivaux, Sets of integers determined by operator-theoretical properties: Jamison and Kazhdan sets in the group ℤ, in Actes du 1-er Congrès National de la SMF—Tours, 2016, Société Mathématique de France, Paris, 2017, pp. 37–75.
C. Badea and S. Grivaux, Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences, Comment. Math. Helv. 95 (2020), 99–127.
G. Barbieri, D. Dikranjan, C. Milan and H. Weber, Answer to Raczkowski’s questions on convergent sequences of integers, Topology Appl. 132 (2003), 89–101.
B. Bekka, P. de la Harpe and A. Valette, Kazhdan’s Property (T), Cambridge University Press, Cambridge, 2008.
V. Bergelson, A. del Junco, M. Lemańczyk and J. Rosenblatt, Rigidity and non-recurrence along sequences, Ergodic Theory Dynam. Systems 34 (2014), 1464–1502.
M. Boshernitzan, Homogeneously distributed sequences and Poincaré sequences of integers of sublacunary growth, Monatsh. Math. 96 (1983), 173–181.
J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math. 61 (1988), 39–72.
I. Chatterji, D. Witte Morris and R. Shah, Relative property (T) for nilpotent subgroups, Documenta Math. 23 (2018), 353–382.
J. R. Choksi and M. G. Nadkarni, Genericity of certain classes of unitary and self-adjoint operators, Canad. Math. Bull. 41 (1998), 137–139.
R. Di Santo, D. Dikranjan and A. Giordano Bruno, Characterized subgroups of the circle group, Ricerche Mat. 67 (2018), 625–655.
H. G. Eggleston, Sets of fractional dimensions which occur in some problems of number theory, Proc. London Math. Soc. (2) 54 (1952), 4–93.
T. Eisner and S. Grivaux, Hilbertian Jamison sequences and rigid dynamical systems, J. Funct. Anal. 261 (2011), 2013–2052.
P. Erdős and A. Rényi, Additive properties of random sequences of positive integers, Acta Arith. 6 (1960), 83–110.
P. Erdős and S. Taylor, On the set of points of convergence of a lacunary trigonometric series and the equidistribution properties of related sequences, Proc. London Math. Soc. (3) 7 (1957), 598–615.
A. Fan and D. Schneider, Recurrence properties of sequences of integers, Sci. China Math. 53 (2010), 641–656.
B. Fayad and A. Kanigowski, Rigidity times for a weakly mixing dynamical system which are not rigidity times for any irrational rotation, Ergodic Theory Dynam. Systems 35 (2015), 2529–2534.
B. Fayad and J.-P. Thouvenot, On the convergence to 0 of mnξ mod 1, Acta Arith. 165 (2014), 327–332.
N. Frantzikinakis, Equidistribution of sparse sequences on nilmanifolds, J. Anal. Math. 109 (2009), 353–395.
N. Frantzikinakis, E. Lesigne and M. Wierdl, Random differences in Szemerédi’s theorem and related results, J. Anal. Math. 130 (2016), 91–133.
H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, NJ, 1981.
H. Furstenberg, Poincaré recurrence and number theory, Bull. Amer. Math. Soc. (N.S.) 5 (1981), 211–234.
H. Furstenberg and B. Weiss, The finite multipliers of infinite ergodic transformations, in The Structure of Attractors in Dynamical Systems, Springer, Berlin-Heidelberg, 1978, pp. 127–132.
M. Goldstern, J. Schmeling and R. Winkler, Metric, fractal dimensional and Baire results on the distribution of subsequences, Math. Nachr. 219 (2000), 97–108.
M. Graev, Free topological groups, Izv. Akad. Nauk SSSR, Ser. Matem. 12 (1948), 278–324.
C. C. Graham and O. C. McGehee, Essays in Commutative Harmonic Analysis, Springer, New York, 1979.
J. Griesmer, Recurrence, rigidity, and popular differences, Ergodic Theory Dynam. Systems 39 (2019), 1299–1316.
S. Grivaux, IP-Dirichlet measures and IP-rigid dynamical systems: an approach via generalized Riesz products, Studia Math. 215 (2013), 237–259.
Y. Katznelson, Sequences of integers dense in the Bohr group, Proc. Royal Inst. Tech. Stockholm (1973), 73–86.
R. Kaufman, Continuous measures and analytic sets, Colloq. Math. 58 (1989), 17–21.
A. Kechris, Classical Descriptive Set Theory, Springer, New York, 1995.
A. Kechris and A. Louveau, Descriptive Set Theory and the Structure of Sets of Uniqueness, Cambridge University Press, Cambridge, 1987.
L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, John Wiley & Sons, New York, 1974.
D. Lenz and P. Stollmann, Generic sets in spaces of measures and generic singular continuous spectrum for Delone Hamiltonians, Duke Math. J. 131 (2006), 203–217.
D. Li, H. Queffélec and L. Rodriguez-Piazza, Some new thin sets of integers in harmonic analysis, J. Anal. Math. 86 (2002), 105–138.
R. Lyons, On measures simultaneously 2- and 3-invariant, Israel J. Math. 61 (1988), 219–224.
R. E. Megginson, An Introduction to Banach Space Theory, Springer, New York, 1998.
M. G. Nadkarni, Spectral Theory of Dynamical Systems, Hindustan Book Agency, New Delhi, 2011.
M. B. Nathanson, Geometric group theory and arithmetic diameter, Publ. Math. Debrecen 79 (2011), 563–572.
S. Neuwirth, Two random constructions inside lacunary sets, Ann. Inst. Fourier 49 (1999), 1853–1867.
J. Nienhuys, Construction of group topologies on Abelian groups, Fund. Math. 75 (1972), 101–116.
J. Nienhuys, Some examples of monothetic groups, Fund. Math. 88 (1975), 163–171.
W. Rudin, Fourier Analysis on Groups, John Wiley & Sons, New York, 1990.
I. Ruzsa, Arithmetical topology, in Number Theory, Vol. I (Budapest, 1987), North-Holland, Amsterdam, 1987, pp. 473–504.
S. Saeki, Bohr compactification and continuous measures, Proc. Amer. Math. Soc. 80 (1980), 244–246.
S. Shkarin, Universal elements for non-linear operators and their applications, J. Math. Anal. Appl. 348 (2008), 193–210.
P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982.
E. Zelenyuk and V. Protasov, Topologies on abelian groups (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), 1090–1107; English translation: Math. USSR-Izv. 37 (1991), 445–460.
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This work was supported in part by the project FRONT of the French National Research Agency (grant ANR-17-CE40-0021) and by the Labex CEMPI (ANR-11-LABX-0007-01).
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Badea, C., Grivaux, S. & Matheron, É. Rigidity sequences, Kazhdan sets and group topologies on the integers. JAMA 143, 313–347 (2021). https://doi.org/10.1007/s11854-021-0165-4
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DOI: https://doi.org/10.1007/s11854-021-0165-4