August  2021, 41(8): 3709-3724. doi: 10.3934/dcds.2021013

Proximality of multidimensional $ \mathscr{B} $-free systems

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University in Toruń, 12/18 Chopin Street, 87-100 Toruń, Poland

 

Received  December 2019 Revised  October 2020 Published  August 2021 Early access  January 2021

Fund Project: The author acknowledges the support of the National Science Centre (NCN), Poland, the doctoral scholarship Etiuda no. 2018/28/T/ST1/00435

We characterize proximality of multidimensional $ \mathscr{B} $-free systems in the case of number fields and lattices in $ \mathbb{Z}^m $, $ m\geq2 $.

Citation: Aurelia Dymek. Proximality of multidimensional $ \mathscr{B} $-free systems. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3709-3724. doi: 10.3934/dcds.2021013
References:
[1]

E. H. El AbdalaouiM. Lemańczyk and T. de la Rue, A dynamical point of view on the set of $\mathscr{B}$-free integers, Int. Math. Res. Not. IMRN, 2015 (2015), 7258-7286.  doi: 10.1093/imrn/rnu164.

[2]

E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433.  doi: 10.1088/0951-7715/16/4/313.

[3]

M. Baake, Solution of the coincidence problem in dimensions $d\leq4$, The mathematics of long-range aperiodic order (Waterloo, ON, 1995), 9–44, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 489, Kluwer Acad. Publ., Dordrecht, 1997.

[4]

M. Baake and C. Huck, Ergodic properties of visible lattice points, Proc. Steklov Inst. Math., 288 (2015), 165-188.  doi: 10.1134/S0081543815010137.

[5]

M. BaakeC. Huck and N. Strungaru, On weak model sets of extremal density, Indag. Math. (N.S.), 28 (2017), 3-31.  doi: 10.1016/j.indag.2016.11.002.

[6]

M. BeiglböckV. Bergelson and A. Fish, Sumset phenomenon in countable amenable groups, Adv. Math., 223 (2010), 416-432.  doi: 10.1016/j.aim.2009.08.009.

[7]

M. BorodzikD. Nguyenand and S. Robins, Tiling the integer lattice with translated sublattices, Mosc. J. Comb. Number Theory, 6 (2016), 3-26. 

[8]

F. Cellarosi and I. Vinogradov, Ergodic properties of $k$-free integers in number fields, J. Mod. Dyn., 7 (2013), 461-488.  doi: 10.3934/jmd.2013.7.461.

[9]

J. P. Clay, Proximality relations in transformation groups, Trans. Amer. Math. Soc., 108 (1963), 88-96.  doi: 10.1090/S0002-9947-1963-0154269-3.

[10]

A. DymekS. KasjanJ. Kułaga-Przymus and M. Lemańczyk, $\mathscr{B}$-free sets and dynamics, Trans. Amer. Math. Soc., 370 (2018), 5425-5489.  doi: 10.1090/tran/7132.

[11]

E. Følner, Generalization of a theorem of Bogoliouboff to topological abelian groups, Math. Scand., 2 (1954), 5–18. https://www.mscand.dk/article/view/10389.

[12]

E. Følner, On groups with full Banach mean value, Math. Scand., 3 (1955), 243-254.  doi: 10.7146/math.scand.a-10442.

[13]

C. Huck, On the logarithmic probability that a random integral ideal is $\mathcal{A}$-free, Ergodic Theory and Dynamical Systems in Their Interactions with Arithmetics and Combinatorics, 249–258, Lecture Notes in Math., 2213, Springer, Cham, 2018.

[14]

S. KasjanG. Keller and M. Lemańczyk, Dynamics of $\mathscr{B}$-free sets: A view through the window, Int. Math. Res. Not. IMRN, 2019 (2019), 2690-2734.  doi: 10.1093/imrn/rnx196.

[15]

M. Newman, Integral Matrices. Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972.

[16]

P. Oprocha and G. Zhang, Topological aspects of dynamics of pairs, tuples and sets, Recent Progress in General Topology, III, Atlantis Press, Paris, 2014,665-709. doi: 10.2991/978-94-6239-024-9_16.

[17]

P. Sarnak, Three Lectures on the Möbius Function, Randomness and Dynamics, http://publications.ias.edu/sites/default/files/MobiusFunctionsLectures(2).pdf.

[18]

T. S. Wu, Proximal relations in topological dynamics, Proc. Amer. Math. Soc., 16 (1965), 513–514 doi: 10.1090/S0002-9939-1965-0179775-4.

show all references

References:
[1]

E. H. El AbdalaouiM. Lemańczyk and T. de la Rue, A dynamical point of view on the set of $\mathscr{B}$-free integers, Int. Math. Res. Not. IMRN, 2015 (2015), 7258-7286.  doi: 10.1093/imrn/rnu164.

[2]

E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433.  doi: 10.1088/0951-7715/16/4/313.

[3]

M. Baake, Solution of the coincidence problem in dimensions $d\leq4$, The mathematics of long-range aperiodic order (Waterloo, ON, 1995), 9–44, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 489, Kluwer Acad. Publ., Dordrecht, 1997.

[4]

M. Baake and C. Huck, Ergodic properties of visible lattice points, Proc. Steklov Inst. Math., 288 (2015), 165-188.  doi: 10.1134/S0081543815010137.

[5]

M. BaakeC. Huck and N. Strungaru, On weak model sets of extremal density, Indag. Math. (N.S.), 28 (2017), 3-31.  doi: 10.1016/j.indag.2016.11.002.

[6]

M. BeiglböckV. Bergelson and A. Fish, Sumset phenomenon in countable amenable groups, Adv. Math., 223 (2010), 416-432.  doi: 10.1016/j.aim.2009.08.009.

[7]

M. BorodzikD. Nguyenand and S. Robins, Tiling the integer lattice with translated sublattices, Mosc. J. Comb. Number Theory, 6 (2016), 3-26. 

[8]

F. Cellarosi and I. Vinogradov, Ergodic properties of $k$-free integers in number fields, J. Mod. Dyn., 7 (2013), 461-488.  doi: 10.3934/jmd.2013.7.461.

[9]

J. P. Clay, Proximality relations in transformation groups, Trans. Amer. Math. Soc., 108 (1963), 88-96.  doi: 10.1090/S0002-9947-1963-0154269-3.

[10]

A. DymekS. KasjanJ. Kułaga-Przymus and M. Lemańczyk, $\mathscr{B}$-free sets and dynamics, Trans. Amer. Math. Soc., 370 (2018), 5425-5489.  doi: 10.1090/tran/7132.

[11]

E. Følner, Generalization of a theorem of Bogoliouboff to topological abelian groups, Math. Scand., 2 (1954), 5–18. https://www.mscand.dk/article/view/10389.

[12]

E. Følner, On groups with full Banach mean value, Math. Scand., 3 (1955), 243-254.  doi: 10.7146/math.scand.a-10442.

[13]

C. Huck, On the logarithmic probability that a random integral ideal is $\mathcal{A}$-free, Ergodic Theory and Dynamical Systems in Their Interactions with Arithmetics and Combinatorics, 249–258, Lecture Notes in Math., 2213, Springer, Cham, 2018.

[14]

S. KasjanG. Keller and M. Lemańczyk, Dynamics of $\mathscr{B}$-free sets: A view through the window, Int. Math. Res. Not. IMRN, 2019 (2019), 2690-2734.  doi: 10.1093/imrn/rnx196.

[15]

M. Newman, Integral Matrices. Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972.

[16]

P. Oprocha and G. Zhang, Topological aspects of dynamics of pairs, tuples and sets, Recent Progress in General Topology, III, Atlantis Press, Paris, 2014,665-709. doi: 10.2991/978-94-6239-024-9_16.

[17]

P. Sarnak, Three Lectures on the Möbius Function, Randomness and Dynamics, http://publications.ias.edu/sites/default/files/MobiusFunctionsLectures(2).pdf.

[18]

T. S. Wu, Proximal relations in topological dynamics, Proc. Amer. Math. Soc., 16 (1965), 513–514 doi: 10.1090/S0002-9939-1965-0179775-4.

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