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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 |
We characterize proximality of multidimensional $ \mathscr{B} $-free systems in the case of number fields and lattices in $ \mathbb{Z}^m $, $ m\geq2 $.
References:
[1] |
E. H. El Abdalaoui, M. 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. Baake, C. 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. Beiglböck, V. 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. Borodzik, D. 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. Dymek, S. Kasjan, J. 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. Kasjan, G. 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 Abdalaoui, M. 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. Baake, C. 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. Beiglböck, V. 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. Borodzik, D. 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. Dymek, S. Kasjan, J. 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. Kasjan, G. 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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