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Transactions of the American Mathematical Society

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$ \mathscr{B}$-free sets and dynamics


Authors: Aurelia Dymek, Stanisław Kasjan, Joanna Kułaga-Przymus and Mariusz Lemańczyk
Journal: Trans. Amer. Math. Soc. 370 (2018), 5425-5489
MSC (2010): Primary 37A35, 37A45, 37B05, 37B10, 37B40; Secondary 11N25
DOI: https://doi.org/10.1090/tran/7132
Published electronically: April 17, 2018
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Abstract: Given $ \mathscr {B}\subset \mathbb{N}$, let $ \eta =\eta _{\mathscr {B}}\in \{0,1\}^{\mathbb{Z}}$ be the characteristic function of the set $ \mathcal {F}_\mathscr {B}:=\mathbb{Z}\setminus \bigcup _{b\in \mathscr {B}}b\mathbb{Z}$ of $ \mathscr {B}$-free numbers. The $ \mathscr {B}$-free shift $ (X_\eta ,S)$, its hereditary closure $ (\widetilde {X}_\eta ,S)$, and (still larger) the $ \mathscr {B}$-admissible shift $ (X_{\mathscr {B}},S)$ are examined. Originated by Sarnak in 2010 for $ \mathscr {B}$ being the set of square-free numbers, the dynamics of $ \mathscr {B}$-free shifts was discussed by several authors for $ \mathscr {B}$ being Erdös; i.e., when $ \mathscr {B}$ is infinite, its elements are pairwise coprime, and $ \sum _{b\in \mathscr {B}}1/b<\infty $: in the Erdös case, we have $ X_\eta =\widetilde {X}_\eta =X_{\mathscr {B}}$.

It is proved that $ X_\eta $ has a unique minimal subset, which turns out to be a Toeplitz dynamical system. Furthermore, a $ \mathscr {B}$-free shift is proximal if and only if $ \mathscr {B}$ contains an infinite coprime subset. It is also shown that for $ \mathscr {B}$ with light tails, i.e., $ \overline {d}(\sum _{b>K}b\mathbb{Z})\to 0$ as $ K\to \infty $, proximality is the same as heredity.

For each $ \mathscr {B}$, it is shown that $ \eta $ is a quasi-generic point for some natural $ S$-invariant measure $ \nu _\eta $ on $ X_\eta $. A special role is played by subshifts given by $ \mathscr {B}$ which are taut, i.e., when $ \boldsymbol {\delta }(\mathcal {F}_{\mathscr {B}})<\boldsymbol {\delta }(\mathcal {F}_{\mathscr {B}\setminus \{b\}})$ for each $ b\in \mathscr {B}$ ( $ \boldsymbol {\delta }$ stands for the logarithmic density). The taut class contains the light tail case; hence all Erdös sets and a characterization of taut sets $ \mathscr {B}$ in terms of the support of $ \nu _\eta $ are given. Moreover, for any $ \mathscr {B}$ there exists a taut $ \mathscr {B}'$ with $ \nu _{\eta _{\mathscr {B}}}=\nu _{\eta _{\mathscr {B}'}}$. For taut sets $ \mathscr {B},\mathscr {B}'$, it holds that $ X_\mathscr {B}=X_{\mathscr {B}'}$ if and only if $ \mathscr {B}=\mathscr {B}'$.

For each $ \mathscr {B}$, it is proved that there exists a taut $ \mathscr {B}'$ such that $ (\widetilde {X}_{\eta _{\mathscr {B}'}},S)$ is a subsystem of $ (\widetilde {X}_{\eta _{\mathscr {B}}},S)$ and $ \widetilde {X}_{\eta _{\mathscr {B}'}}$ is a quasi-attractor. In particular, all invariant measures for $ (\widetilde {X}_{\eta _{\mathscr {B}}},S)$ are supported by $ \widetilde {X}_{\eta _{\mathscr {B}'}}$. Moreover, the system $ (\widetilde {X}_\eta ,S)$ is shown to be intrinsically ergodic for an arbitrary $ \mathscr {B}$. A description of all probability invariant measures for $ (\widetilde {X}_\eta ,S)$ is given. The topological entropies of $ (\widetilde {X}_\eta ,S)$ and $ (X_\mathscr {B},S)$ are shown to be the same and equal to $ \overline {d}(\mathcal {F}_\mathscr {B})$ ( $ \overline {d}$ stands for the upper density).

Finally, some applications in number theory on gaps between consecutive $ \mathscr {B}$-free numbers are given, and some of these results are applied to the set of abundant numbers.


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Additional Information

Aurelia Dymek
Affiliation: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
Email: aurbart@mat.umk.pl

Stanisław Kasjan
Affiliation: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
Email: skasjan@mat.umk.pl

Joanna Kułaga-Przymus
Affiliation: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Wars- zawa, Poland–and–Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
Email: joanna.kulaga@gmail.com

Mariusz Lemańczyk
Affiliation: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
Email: mlem@mat.umk.pl

DOI: https://doi.org/10.1090/tran/7132
Received by editor(s): November 6, 2015
Received by editor(s) in revised form: June 3, 2016, and November 10, 2016
Published electronically: April 17, 2018
Additional Notes: This research was supported by Narodowe Centrum Nauki grant UMO-2014/15/B/ST1/03736
Article copyright: © Copyright 2018 American Mathematical Society

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