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October  2019, 39(10): 5891-5921. doi: 10.3934/dcds.2019258

Multiplicative combinatorial properties of return time sets in minimal dynamical systems

1. 

Department of Mathematics, Northeastern University, Boston, MA, USA

2. 

Department of Mathematics, The Ohio State University, Columbus, OH, USA

3. 

Department of Mathematics, Northwestern University, Evanston, IL, USA

Received  October 2018 Revised  March 2019 Published  July 2019

We investigate the relationship between the dynamical properties of minimal topological dynamical systems and the multiplicative combinatorial properties of return time sets arising from those systems. In particular, we prove that for a residual set of points in any minimal system, the set of return times to any non-empty, open set contains arbitrarily long geometric progressions. Under the separate assumptions of total minimality and distality, we prove that return time sets have positive multiplicative upper Banach density along $ \mathbb{N} $ and along cosets of multiplicative subsemigroups of $ \mathbb{N} $, respectively. The primary motivation for this work is the long-standing open question of whether or not syndetic subsets of the positive integers contain arbitrarily long geometric progressions; our main result is some evidence for an affirmative answer to this question.

Citation: Daniel Glasscock, Andreas Koutsogiannis, Florian Karl Richter. Multiplicative combinatorial properties of return time sets in minimal dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5891-5921. doi: 10.3934/dcds.2019258
References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide, 3rd edition, Springer, Berlin, 2006. Google Scholar

[2]

M. BeiglböckV. BergelsonN. Hindman and D. Strauss, Multiplicative structures in additively large sets, J. Combin. Theory Ser. A, 113 (2006), 1219-1242. doi: 10.1016/j.jcta.2005.11.003. Google Scholar

[3]

V. Bergelson, Multiplicatively large sets and ergodic Ramsey theory, Israel J. Math., 148 (2005), 23–40, Probability in mathematics. doi: 10.1007/BF02775431. Google Scholar

[4]

V. BergelsonJ. C. ChristophersonD. Robertson and P. Zorin-Kranich, Finite products sets and minimally almost periodic groups, J. Funct. Anal., 270 (2016), 2126-2167. doi: 10.1016/j.jfa.2015.12.008. Google Scholar

[5]

V. Bergelson and D. Glasscock, On the interplay between notions of additive and multiplicative largeness and its combinatorial applications, URL http://arXiv.org/abs/1610.09771.Google Scholar

[6]

V. Bergelson and A. Leibman, IPr*-recurrence and nilsystems, Adv. Math., 339 (2018), 642-656. doi: 10.1016/j.aim.2018.09.032. Google Scholar

[7] M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755316. Google Scholar
[8] T. Downarowicz, Entropy in Dynamical Systems, vol. 18 of New Mathematical Monographs, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511976155. Google Scholar
[9]

M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, vol. 259 of Graduate Texts in Mathematics, Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2. Google Scholar

[10]

M. K. Fort Jr., Points of continuity of semi-continuous functions, Publ. Math. Debrecen, 2 (1951), 100-102. Google Scholar

[11]

H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515. doi: 10.2307/2373137. Google Scholar

[12]

H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math., 31 (1977), 204-256. doi: 10.1007/BF02813304. Google Scholar

[13]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981, M. B. Porter Lectures. doi: 10.1007/BF02775431. Google Scholar

[14]

H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for IP-systems and combinatorial theory, J. Analyse Math., 45 (1985), 117-168. doi: 10.1007/BF02792547. Google Scholar

[15]

H. Furstenberg and B. Weiss, Topological dynamics and combinatorial number theory, J. Analyse Math., 34 (1978), 61–85 (1979). doi: 10.1007/BF02790008. Google Scholar

[16]

E. Glasner, Topological ergodic decompositions and applications to products of powers of a minimal transformation, J. Anal. Math., 64 (1994), 241-262. doi: 10.1007/BF03008411. Google Scholar

[17]

E. Glasner, Structure theory as a tool in topological dynamics, in Descriptive Set Theory and Dynamical Systems (Marseille-Luminy, 1996), London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 277 (2000), 173–209. Google Scholar

[18]

S. Kakeya and S. Morimoto, On a theorem of mm. bandet and van der waerden, Japanese journal of mathematics: transactions and abstracts, 7 (1930), 163-165. doi: 10.4099/jjm1924.7.0_163. Google Scholar

[19]

S. KolyadaL. Snoha and S. Trofimchuk, Noninvertible minimal maps, Fund. Math., 168 (2001), 141-163. doi: 10.4064/fm168-2-5. Google Scholar

[20]

J. Moreira, Monochromatic sums and products in $\mathbb{N}$, Ann. of Math. (2), 185 (2017), 1069-1090. doi: 10.4007/annals.2017.185.3.10. Google Scholar

[21]

B. R. Patil, Geometric progressions in syndetic sets, To appear in Archiv der Mathematik, URL http://arXiv.org/abs/1808.09230.Google Scholar

[22]

R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, vol. 1364 of Lecture Notes in Mathematics, 2nd edition, Springer-Verlag, Berlin, 1993. Google Scholar

[23]

E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Acta Arith., 27 (1975), 199–245, Collection of articles in memory of Juriĭ Vladimirovič Linnik. doi: 10.4064/aa-27-1-199-245. Google Scholar

[24]

B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wiskd., Ⅱ. Ser., 15 (1927), 212-216. Google Scholar

[25]

X. Ye, D-function of a minimal set and an extension of sharkovskii's theorem to minimal sets, Ergodic Theory and Dynamical Systems, 12 (1992), 365-376. doi: 10.1017/S0143385700006817. Google Scholar

show all references

References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide, 3rd edition, Springer, Berlin, 2006. Google Scholar

[2]

M. BeiglböckV. BergelsonN. Hindman and D. Strauss, Multiplicative structures in additively large sets, J. Combin. Theory Ser. A, 113 (2006), 1219-1242. doi: 10.1016/j.jcta.2005.11.003. Google Scholar

[3]

V. Bergelson, Multiplicatively large sets and ergodic Ramsey theory, Israel J. Math., 148 (2005), 23–40, Probability in mathematics. doi: 10.1007/BF02775431. Google Scholar

[4]

V. BergelsonJ. C. ChristophersonD. Robertson and P. Zorin-Kranich, Finite products sets and minimally almost periodic groups, J. Funct. Anal., 270 (2016), 2126-2167. doi: 10.1016/j.jfa.2015.12.008. Google Scholar

[5]

V. Bergelson and D. Glasscock, On the interplay between notions of additive and multiplicative largeness and its combinatorial applications, URL http://arXiv.org/abs/1610.09771.Google Scholar

[6]

V. Bergelson and A. Leibman, IPr*-recurrence and nilsystems, Adv. Math., 339 (2018), 642-656. doi: 10.1016/j.aim.2018.09.032. Google Scholar

[7] M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755316. Google Scholar
[8] T. Downarowicz, Entropy in Dynamical Systems, vol. 18 of New Mathematical Monographs, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511976155. Google Scholar
[9]

M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, vol. 259 of Graduate Texts in Mathematics, Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2. Google Scholar

[10]

M. K. Fort Jr., Points of continuity of semi-continuous functions, Publ. Math. Debrecen, 2 (1951), 100-102. Google Scholar

[11]

H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515. doi: 10.2307/2373137. Google Scholar

[12]

H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math., 31 (1977), 204-256. doi: 10.1007/BF02813304. Google Scholar

[13]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981, M. B. Porter Lectures. doi: 10.1007/BF02775431. Google Scholar

[14]

H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for IP-systems and combinatorial theory, J. Analyse Math., 45 (1985), 117-168. doi: 10.1007/BF02792547. Google Scholar

[15]

H. Furstenberg and B. Weiss, Topological dynamics and combinatorial number theory, J. Analyse Math., 34 (1978), 61–85 (1979). doi: 10.1007/BF02790008. Google Scholar

[16]

E. Glasner, Topological ergodic decompositions and applications to products of powers of a minimal transformation, J. Anal. Math., 64 (1994), 241-262. doi: 10.1007/BF03008411. Google Scholar

[17]

E. Glasner, Structure theory as a tool in topological dynamics, in Descriptive Set Theory and Dynamical Systems (Marseille-Luminy, 1996), London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 277 (2000), 173–209. Google Scholar

[18]

S. Kakeya and S. Morimoto, On a theorem of mm. bandet and van der waerden, Japanese journal of mathematics: transactions and abstracts, 7 (1930), 163-165. doi: 10.4099/jjm1924.7.0_163. Google Scholar

[19]

S. KolyadaL. Snoha and S. Trofimchuk, Noninvertible minimal maps, Fund. Math., 168 (2001), 141-163. doi: 10.4064/fm168-2-5. Google Scholar

[20]

J. Moreira, Monochromatic sums and products in $\mathbb{N}$, Ann. of Math. (2), 185 (2017), 1069-1090. doi: 10.4007/annals.2017.185.3.10. Google Scholar

[21]

B. R. Patil, Geometric progressions in syndetic sets, To appear in Archiv der Mathematik, URL http://arXiv.org/abs/1808.09230.Google Scholar

[22]

R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, vol. 1364 of Lecture Notes in Mathematics, 2nd edition, Springer-Verlag, Berlin, 1993. Google Scholar

[23]

E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Acta Arith., 27 (1975), 199–245, Collection of articles in memory of Juriĭ Vladimirovič Linnik. doi: 10.4064/aa-27-1-199-245. Google Scholar

[24]

B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wiskd., Ⅱ. Ser., 15 (1927), 212-216. Google Scholar

[25]

X. Ye, D-function of a minimal set and an extension of sharkovskii's theorem to minimal sets, Ergodic Theory and Dynamical Systems, 12 (1992), 365-376. doi: 10.1017/S0143385700006817. Google Scholar

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