On Katznelson’s Question for skew-product systems

Glasscock, Daniel; Koutsogiannis, Andreas; Richter, Florian

doi:10.1090/bull/1764

On Katznelson’s Question for skew-product systems
HTML articles powered by AMS MathViewer

by Daniel Glasscock, Andreas Koutsogiannis and Florian K. Richter;

Bull. Amer. Math. Soc. 59 (2022), 569-606

DOI: https://doi.org/10.1090/bull/1764

Published electronically: August 4, 2022

HTML | PDF | Request permission

Abstract:

Katznelson’s Question is a long-standing open question concerning recurrence in topological dynamics with strong historical and mathematical ties to open problems in combinatorics and harmonic analysis. In this article, we give a positive answer to Katznelson’s Question for certain towers of skew-product extensions of equicontinuous systems, including systems of the form $(x,t) \mapsto (x + \alpha , t + h(x))$. We describe which frequencies must be controlled for in order to ensure recurrence in such systems, and we derive combinatorial corollaries concerning the difference sets of syndetic subsets of the natural numbers.

References

Vitaly Bergelson, Hillel Furstenberg, and Benjamin Weiss, Piecewise-Bohr sets of integers and combinatorial number theory, Topics in discrete mathematics, Algorithms Combin., vol. 26, Springer, Berlin, 2006, pp. 13–37. MR 2249261, DOI 10.1007/3-540-33700-8_{2}
Michael Boshernitzan and Eli Glasner, On two recurrence problems, Fund. Math. 206 (2009), 113–130. MR 2576263, DOI 10.4064/fm206-0-7
Michael Björklund and John T. Griesmer, Bohr sets in triple products of large sets in amenable groups, J. Fourier Anal. Appl. 25 (2019), no. 3, 923–936. MR 3953491, DOI 10.1007/s00041-018-9615-5
Vitaly Bergelson and Daniel Glasscock, On the interplay between additive and multiplicative largeness and its combinatorial applications, J. Combin. Theory Ser. A 172 (2020), 105203, 60. MR 4050733, DOI 10.1016/j.jcta.2019.105203
Nikolay N. Bogolyubov, Some algebraical properties of almost periods (in Russian), Zap. kafedry mat. fiziki Kiev, 4 (1939), 185–194.
Vitaly Bergelson and Imre Z. Ruzsa, Sumsets in difference sets, Israel J. Math. 174 (2009), 1–18. MR 2581205, DOI 10.1007/s11856-009-0100-3
Yair Caro, Adriana Hansberg, and Amanda Montejano, Zero-sum subsequences in bounded-sum $\{-1,1\}$-sequences, J. Combin. Theory Ser. A 161 (2019), 387–419. MR 3861785, DOI 10.1016/j.jcta.2018.09.001
Robert Ellis and Harvey B. Keynes, Bohr compactifications and a result of Følner, Israel J. Math. 12 (1972), 314–330. MR 313438, DOI 10.1007/BF02790758
Nikos Frantzikinakis and Randall McCutcheon, Ergodic theory: recurrence, Mathematics of complexity and dynamical systems. Vols. 1–3, Springer, New York, 2012, pp. 357–368. MR 3220681, DOI 10.1007/978-1-4614-1806-1_{2}3
Erling Følner, Almost periodic functions on Abelian groups, C. R. Dixième Congrès Math. Scandinaves 1946, Gjellerup, Copenhagen, 1947, pp. 356–362. MR 19094
Erling Følner, A proof of the main theorem for almost periodic functions in an Abelian group, Ann. of Math. (2) 50 (1949), 559–569. MR 30954, DOI 10.2307/1969549
Erling Følner, Generalization of a theorem of Bogolioùboff to topological abelian groups. With an appendix on Banach mean values in non-abelian groups, Math. Scand. 2 (1954), 5–18. MR 64062, DOI 10.7146/math.scand.a-10389
Erling Følner, Note on a generalization of a theorem of Bogolioùboff, Math. Scand. 2 (1954), 224–226. MR 69188, DOI 10.7146/math.scand.a-10408
H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, NJ, 1981. M. B. Porter Lectures. MR 603625, DOI 10.1515/9781400855162
Walter Helbig Gottschalk and Gustav Arnold Hedlund, Topological dynamics, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, RI, 1955. MR 74810, DOI 10.1090/coll/036
Eli Glasner, On minimal actions of Polish groups, Topology Appl. 85 (1998), no. 1-3, 119–125. 8th Prague Topological Symposium on General Topology and Its Relations to Modern Analysis and Algebra (1996). MR 1617456, DOI 10.1016/S0166-8641(97)00143-0
Eli Glasner, Structure theory as a tool in topological dynamics, Descriptive set theory and dynamical systems (Marseille-Luminy, 1996) London Math. Soc. Lecture Note Ser., vol. 277, Cambridge Univ. Press, Cambridge, 2000, pp. 173–209. MR 1774426
Alfred Geroldinger and Imre Z. Ruzsa, Combinatorial number theory and additive group theory, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2009. Courses and seminars from the DocCourse in Combinatorics and Geometry held in Barcelona, 2008. MR 2547479, DOI 10.1007/978-3-7643-8962-8
Sophie Grivaux and Maria Roginskaya, Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle, Czechoslovak Math. J. 63(138) (2013), no. 3, 603–627. MR 3125645, DOI 10.1007/s10587-013-0043-z
John T. Griesmer, Sumsets of dense sets and sparse sets, Israel J. Math. 190 (2012), 229–252. MR 2956240, DOI 10.1007/s11856-012-0008-1
John T. Griesmer, Separating Bohr denseness from measurable recurrence, Discrete Anal. , posted on (2021), Paper No. 9, 20. MR 4308022, DOI 10.19086/da
Bernard Host, Bryna Kra, and Alejandro Maass, Variations on topological recurrence, Monatsh. Math. 179 (2016), no. 1, 57–89. MR 3439271, DOI 10.1007/s00605-015-0765-0
P. Hellekalek and G. Larcher, On Weyl sums and skew products over irrational rotations, Theoret. Comput. Sci. 65 (1989), no. 2, 189–196. MR 1020486, DOI 10.1016/0304-3975(89)90043-1
Norbert Hegyvári and Imre Z. Ruzsa, Additive structure of difference sets and a theorem of Følner, Australas. J. Combin. 64 (2016), 437–443. MR 3457812
Wen Huang, Song Shao, and Xiangdong Ye, Nil Bohr-sets and almost automorphy of higher order, Mem. Amer. Math. Soc. 241 (2016), no. 1143, v+83. MR 3476203, DOI 10.1090/memo/1143
Y. Katznelson, Chromatic numbers of Cayley graphs on $\Bbb Z$ and recurrence, Combinatorica 21 (2001), no. 2, 211–219. Paul Erdős and his mathematics (Budapest, 1999). MR 1832446, DOI 10.1007/s004930100019
A. Ya. Khintchine, Continued fractions, P. Noordhoff Ltd., Groningen, 1963. Translated by Peter Wynn. MR 161834
Igor Kříž, Large independent sets in shift-invariant graphs: solution of Bergelson’s problem, Graphs Combin. 3 (1987), no. 2, 145–158. MR 932131, DOI 10.1007/BF01788538
Magnus B. Landstad, On the Bohr topology in amenable topological groups, Math. Scand. 28 (1971), 207–214. MR 304548, DOI 10.7146/math.scand.a-11017
Anh N. Le and Thái Hoàng Lê, Bohr sets in sumsets I: Compact groups, arXiv:2112.11997, 2021.
Douglas C. McMahon, Relativized weak disjointness and relatively invariant measures, Trans. Amer. Math. Soc. 236 (1978), 225–237. MR 467704, DOI 10.1090/S0002-9947-1978-0467704-9
Yuval Peres, Application of Banach limits to the study of sets of integers, Israel J. Math. 62 (1988), no. 1, 17–31. MR 947826, DOI 10.1007/BF02767350
Vladimir Pestov, Chapter 44—Forty-plus annotated questions about large topological groups, In Elliott Pearl, editor, Open Problems in Topology II, pages 439–450. Elsevier, Amsterdam, 2007.
Karl Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, vol. 2, Cambridge University Press, Cambridge, 1989. Corrected reprint of the 1983 original. MR 1073173
Imre Z. Ruzsa, Uniform distribution, positive trigonometric polynomials and difference sets, Seminar on Number Theory, 1981/1982, Univ. Bordeaux I, Talence, 1982, pp. Exp. No. 18, 18. MR 695335
Imre Z. Ruzsa, Difference sets and the bohr topology, Unpublished manuscript, 1985.
Hugo Steinhaus, Sur les distances des points dans les ensembles de mesure positive, Fundamenta Mathematicae, 1 (1920), no. 1, 93–104.
William A. Veech, The equicontinuous structure relation for minimal Abelian transformation groups, Amer. J. Math. 90 (1968), 723–732. MR 232377, DOI 10.2307/2373480
André Weil, L’intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 869, Hermann & Cie, Paris, 1940 (French). [This book has been republished by the author at Princeton, N. J., 1941.]. MR 5741
Benjamin Weiss, Single orbit dynamics, CBMS Regional Conference Series in Mathematics, vol. 95, American Mathematical Society, Providence, RI, 2000. MR 1727510, DOI 10.1090/cbms/095