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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 and Continuous Dynamical Systems, 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.

[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.

[3]

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

[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.

[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.

[6]

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

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

[10]

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

[11]

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

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[19]

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

[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.

[21]

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

[22]

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

[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.

[24]

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

[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.

show all references

References:
[1]

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

[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.

[3]

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

[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.

[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.

[6]

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

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

[10]

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

[11]

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

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[19]

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

[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.

[21]

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

[22]

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

[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.

[24]

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

[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.

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