ℬ-free sets and dynamics
Authors: Dymek Aurelia , Kasjan Stanisław , Kułaga-Przymus Joanna , Lemańczyk Mariusz 
Published: July 2017

Given

B

⊂

N

\mathscr {B}\subset \mathbb {N}

, let

η
=

η

B

∈
{
0
,
1

}

Z

\eta =\eta _{\mathscr {B}}\in \{0,1\}^{\mathbb {Z}}

be the characteristic function of the set

F

B

:=

Z

∖

⋃

b
∈

B

b

Z

\mathcal {F}_\mathscr {B}:=\mathbb {Z}\setminus \bigcup _{b\in \mathscr {B}}b\mathbb {Z}

of

B

\mathscr {B}

-free numbers. The

B

\mathscr {B}

-free shift

(

X
η

,
S
)

(X_\eta ,S)

, its hereditary closure

(

X
~

η

,
S
)

(\widetilde {X}_\eta ,S)

, and (still larger) the

B

\mathscr {B}

-admissible shift

(

X

B

,
S
)

(X_{\mathscr {B}},S)

are examined. Originated by Sarnak in 2010 for

B

\mathscr {B}

being the set of square-free numbers, the dynamics of

B

\mathscr {B}

-free shifts was discussed by several authors for

B

\mathscr {B}

being Erdös; i.e., when

B

\mathscr {B}

is infinite, its elements are pairwise coprime, and

∑

b
∈

B

1

/

b
>
∞

\sum _{b\in \mathscr {B}}1/b>\infty

: in the Erdös case, we have

X
η

=

X
~

η

=

X

B

X_\eta =\widetilde {X}_\eta =X_{\mathscr {B}}

.

It is proved that

X
η

X_\eta

has a unique minimal subset, which turns out to be a Toeplitz dynamical system. Furthermore, a

B

\mathscr {B}

-free shift is proximal if and only if

B

\mathscr {B}

contains an infinite coprime subset. It is also shown that for

B

\mathscr {B}

with light tails, i.e.,

d
¯

(

∑

b
>
K

b

Z

)
→
0

\overline {d}(\sum _{b>K}b\mathbb {Z})\to 0

as

K
→
∞

K\to \infty

, proximality is the same as heredity.

For each

B

\mathscr {B}

, it is shown that

η
\eta

is a quasi-generic point for some natural

S
S

-invariant measure

ν
η

\nu _\eta

on

X
η

X_\eta

. A special role is played by subshifts given by

B

\mathscr {B}

which are taut, i.e., when

δ
(

F

B

)
>
δ
(

F

B

∖
{
b
}

)

\boldsymbol {\delta }(\mathcal {F}_{\mathscr {B}})>\boldsymbol {\delta }(\mathcal {F}_{\mathscr {B}\setminus \{b\}})

for each

b
∈

B

b\in \mathscr {B}

(

δ
\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

B

\mathscr {B}

in terms of the support of

ν
η

\nu _\eta

are given. Moreover, for any

B

\mathscr {B}

there exists a taut

B

′

\mathscr {B}’

with

ν

η

B

=

ν

η

B

′

\nu _{\eta _{\mathscr {B}}}=\nu _{\eta _{\mathscr {B}’}}

. For taut sets

B

,

B

′

\mathscr {B},\mathscr {B}’

, it holds that

X

B

=

X

B

′

X_\mathscr {B}=X_{\mathscr {B}’}

if and only if

B

=

B

′

\mathscr {B}=\mathscr {B}’

.

For each

B

\mathscr {B}

, it is proved that there exists a taut

B

′

\mathscr {B}’

such that

(

X
~

η

B

′

,
S
)

(\widetilde {X}_{\eta _{\mathscr {B}’}},S)

is a subsystem of

(

X
~

η

B

,
S
)

(\widetilde {X}_{\eta _{\mathscr {B}}},S)

and

X
~

η

B

′

\widetilde {X}_{\eta _{\mathscr {B}’}}

is a quasi-attractor. In particular, all invariant measures for

(

X
~

η

B

,
S
)

(\widetilde {X}_{\eta _{\mathscr {B}}},S)

are supported by

X
~

η

B

′

\widetilde {X}_{\eta _{\mathscr {B}’}}

. Moreover, the system

(

X
~

η

,
S
)

(\widetilde {X}_\eta ,S)

is shown to be intrinsically ergodic for an arbitrary

B

\mathscr {B}

. A description of all probability invariant measures for

(

X
~

η

,
S
)

(\widetilde {X}_\eta ,S)

is given. The topological entropies of

(

X
~

η

,
S
)

(\widetilde {X}_\eta ,S)

and

(

X

B

,
S
)

(X_\mathscr {B},S)

are shown to be the same and equal to

d
¯

(

F

B

)

\overline {d}(\mathcal {F}_\mathscr {B})

(

d
¯

\overline {d}

stands for the upper density).

Finally, some applications in number theory on gaps between consecutive

B

\mathscr {B}

-free numbers are given, and some of these results are applied to the set of abundant numbers.