# American Institute of Mathematical Sciences

• Previous Article
Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity
• DCDS Home
• This Issue
• Next Article
The nonlinear Schrödinger equations with harmonic potential in modulation spaces
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. Beiglböck, V. Bergelson, N. 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. Bergelson, J. C. Christopherson, D. 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. Kolyada, L. 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. Beiglböck, V. Bergelson, N. 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. Bergelson, J. C. Christopherson, D. 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. Kolyada, L. 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
 [1] Caili Sang, Zhen Chen. $E$-eigenvalue localization sets for tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019042 [2] Anna Lenzhen, Babak Modami, Kasra Rafi. Teichmüller geodesics with $d$-dimensional limit sets. Journal of Modern Dynamics, 2018, 12: 261-283. doi: 10.3934/jmd.2018010 [3] Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129 [4] Renaud Leplaideur, Benoît Saussol. Large deviations for return times in non-rectangle sets for axiom a diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 327-344. doi: 10.3934/dcds.2008.22.327 [5] Hongyong Cui, Peter E. Kloeden, Meihua Yang. Forward omega limit sets of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-12. doi: 10.3934/dcdss.2020065 [6] Boju Jiang, Jaume Llibre. Minimal sets of periods for torus maps. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 301-320. doi: 10.3934/dcds.1998.4.301 [7] Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231 [8] Xiaopeng Zhao. Space-time decay estimates of solutions to liquid crystal system in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2019, 18 (1) : 1-13. doi: 10.3934/cpaa.2019001 [9] Edcarlos D. Silva, José Carlos de Albuquerque, Uberlandio Severo. On a class of linearly coupled systems on $\mathbb{R}^N$ involving asymptotically linear terms. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3089-3101. doi: 10.3934/cpaa.2019138 [10] Byungik Kahng, Miguel Mendes. The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems. Conference Publications, 2013, 2013 (special) : 393-406. doi: 10.3934/proc.2013.2013.393 [11] Luiz Felipe Nobili França. Partially hyperbolic sets with a dynamically minimal lamination. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2717-2729. doi: 10.3934/dcds.2018114 [12] Jaume Llibre, Ricardo Miranda Martins, Marco Antonio Teixeira. On the birth of minimal sets for perturbed reversible vector fields. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 763-777. doi: 10.3934/dcds.2011.31.763 [13] Ronald A. Knight. Compact minimal sets in continuous recurrent flows. Conference Publications, 1998, 1998 (Special) : 397-407. doi: 10.3934/proc.1998.1998.397 [14] Gary Froyland, Philip K. Pollett, Robyn M. Stuart. A closing scheme for finding almost-invariant sets in open dynamical systems. Journal of Computational Dynamics, 2014, 1 (1) : 135-162. doi: 10.3934/jcd.2014.1.135 [15] Yujun Ju, Dongkui Ma, Yupan Wang. Topological entropy of free semigroup actions for noncompact sets. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 995-1017. doi: 10.3934/dcds.2019041 [16] Roberta Fabbri, Sylvia Novo, Carmen Núñez, Rafael Obaya. Null controllable sets and reachable sets for nonautonomous linear control systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1069-1094. doi: 10.3934/dcdss.2016042 [17] Teresa Alberico, Costantino Capozzoli, Luigi D'Onofrio, Roberta Schiattarella. $G$-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb R^3$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 129-137. doi: 10.3934/dcdss.2019009 [18] Guoyuan Chen, Yong Liu, Juncheng Wei. Nondegeneracy of harmonic maps from ${{\mathbb{R}}^{2}}$ to ${{\mathbb{S}}^{2}}$. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-19. doi: 10.3934/dcds.2019228 [19] Hiromichi Nakayama, Takeo Noda. Minimal sets and chain recurrent sets of projective flows induced from minimal flows on $3$-manifolds. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 629-638. doi: 10.3934/dcds.2005.12.629 [20] Xi-Nan Ma, Jiang Ye, Yun-Hua Ye. Principal curvature estimates for the level sets of harmonic functions and minimal graphs in $R^3$. Communications on Pure & Applied Analysis, 2011, 10 (1) : 225-243. doi: 10.3934/cpaa.2011.10.225

2018 Impact Factor: 1.143