1 Introduction
1.1 Toeplitz subshifts
Let
$\eta \in \{0,1\}^{\mathbb Z}$
be a non-periodic Toeplitz sequence [Reference Jacobs and Keane17] with period structure
$p_1\mid p_2\mid p_3 \ldots \to \infty $
. That means for each
$k\in {\mathbb Z}$
, there exists
$n\ge 1$
such that
$\eta _{|k+p_n{\mathbb Z}}$
is constant, but
$\eta $
is not periodic (the latter excludes trivial cases). Denote the orbit closure of
$\eta $
under the left shift
$\sigma $
on
$\{0,1\}^{\mathbb Z}$
by
$X_\eta $
. Each such subshift is called a Toeplitz shift. It follows from the work of Williams [Reference Williams25] that such systems are in fact almost 1-1 extensions of their maximal equicontinuous factor (MEF), in this case of an associated odometer
$(G,T)$
, where G is the compact topological group
$\varprojlim {\mathbb Z}/p_n{\mathbb Z}$
built from the period structure
$(p_n)_{n\ge 1}$
, and T is the translation by
$(1, 1,\ldots )$
. Recall that odometers are minimal, equicontinuous and zero-dimensional dynamical systems, and the conjunction of these three properties characterizes them among all topological dynamical systems, see [Reference Downarowicz, Kolyada, Manin and Ward8].
1.2 The centralizer
For any subshift
$(Y,\sigma )$
, that is, a shift invariant closed subset
$Y\subseteq \{0,1\}^{\mathbb Z}$
, the automorphism group (or centralizer) is the group of all homeomorphisms
$U\colon Y\to Y$
which commute with
$\sigma $
. Its elements are sliding block codes [Reference Hedlund16]. Therefore, the automorphism group is countable. Since all powers of the shift are elements of the centralizer, it contains a copy of
${\mathbb Z}$
as a normal subgroup. We say that the automorphism group is trivial if it consists solely of powers of the shift.
Centralizers are studied for various classes of systems. Bułatek and Kwiatkowski [Reference Bułatek and Kwiatkowski2] do it for Toeplitz subshifts with separated holes (Sh) using elements of the associated odometer
$(G,T)$
. Moreover, in [Reference Bułatek and Kwiatkowski3], they deliver examples of Toeplitz subshifts with positive topological entropy and trivial automorphism group. More recently, Cyr and Kra study automorphism groups of subshifts of subquadratic and linear growth in [Reference Cyr and Kra4, Reference Cyr and Kra5]. In [Reference Cyr and Kra5], they prove that the cosets of powers of the shift in the centralizer of any topologically transitive subshift of subquadratic growth form a periodic group. In [Reference Cyr and Kra4], they show that any minimal subshift with non-superlinear complexity has a virtually
${\mathbb Z}$
automorphism group, answering the question asked in [Reference Salo and Törmä24]. In [Reference Donoso, Durand, Maass and Petite6], the same result is shown independently with different methods by Donoso et al. However, the centralizer can be quite a complicated group. In [Reference Salo23], Salo gives an example of a Toeplitz subshift with not finitely generated automorphism group. We provide another example with this property, see equation (53) and Corollary 3.25.
1.3 This paper’s contributions to general Toeplitz shifts
Let
$\eta $
be a Toeplitz sequence with period structure
$(p_n)_{n\ge 1}$
and recall from [Reference Bułatek and Kwiatkowski2] that a position
$k\in {\mathbb Z}$
is called a hole at level N if
$\eta _{|k+p_N{\mathbb Z}}$
is not constant. We refine this concept and call such a hole essential if the residue class
$k+p_N{\mathbb Z}$
contains holes of each level
$n\ge N$
, see Definition 2.2. The minimal period
$\tilde {\tau }_N$
of the set of essential holes at level N divides
$p_N$
and, under a (seemingly strong) additional assumption, Theorem 2.8 provides restrictions on the size of the centralizer in terms of the quotients
$p_N/\tilde {\tau }_N$
. A direct application of this result to a variant of the Garcia–Hedlund sequence is discussed in Example 2.19. After that, using a mixture of topological and arithmetic arguments, we show that the additional assumption is satisfied more often than one may expect—the main tool is Theorem 2.31. Along this way, we exploit suitable topological variants of the separated holes condition (Sh) from [Reference Bułatek and Kwiatkowski2] in Proposition 2.16, and verify these conditions under arithmetic assumptions tailor-made for the
${\mathcal B}$
-free case in Proposition 2.30. We note that our techniques generalize the setting from [Reference Bułatek and Kwiatkowski2], but are kind of transverse to the setting from [Reference Bułatek and Kwiatkowski3], see Remark 2.10.
1.4 The centralizer of
${\mathcal B}$
-free Toeplitz shifts
For any set
${\mathcal B}\subseteq \{2,3,\ldots \}$
, let
$$ \begin{align*} {\mathcal M}_{\mathcal B}:=\bigcup_{b\in{\mathcal B}}b{\mathbb Z} \end{align*} $$
be the set of multiples of
${\mathcal B}$
and
$$ \begin{align*} {\mathcal F}_{\mathcal B}:={\mathbb Z}\setminus{\mathcal M}_{\mathcal B} \end{align*} $$
the set of
${\mathcal B}$
-free numbers. One can easily modify a set
${\mathcal B}$
to have the same set of multiples and to be primitive, that is,
$b\nmid b'$
for different
$b,b'\in {\mathcal B}$
. So, we will tacitly assume that
${\mathcal B}$
is primitive. The investigation of sets of multiples and
${\mathcal B}$
-free numbers has a quite long history, see [Reference Hall15]. Recently, the subshifts associated with
${\mathcal B}$
-free numbers are under intensive study, see e.g. [Reference Dymek, Kasjan, Kułaga-Przymus and Lemańczyk12, Reference Kasjan, Keller and Lemańczyk18] and references therein. Namely, let
$\eta \in \{0,1\}^{\mathbb Z}$
be the characteristic function of the
${\mathcal B}$
-free numbers. Minimality of
$(X_\eta ,\sigma )$
is equivalent to
$\eta $
being Toeplitz is shown in [Reference Kasjan, Keller and Lemańczyk18, Theorem B], whenever
${\mathcal B}$
is taut. The tautness assumption was removed in [Reference Dymek, Kasjan and Kułaga-Przymus11, Theorem 3.7]. So, in the
${\mathcal B}$
-free context,
$(X_\eta ,\sigma )$
is minimal if and only if
$\eta $
is a Toeplitz sequence.
In the case of
${\mathcal B}$
-free Toeplitz subshifts, the previously cited results for low complexity systems are not useful, because even very simple
${\mathcal B}$
-free Toeplitz systems may have superpolynomial complexity, see §3.6, where this is shown for
${\mathcal B}={\mathcal B}_1:=\{2^nc_n:n\in {\mathbb N}\}$
with pairwise coprime odd
$c_n$
. Although the separated holes condition (Sh) is satisfied for this simple example, we were not able to use the results from [Reference Bułatek and Kwiatkowski2] to determine the centralizer. However, its triviality was shown with more direct methods in [Reference Dymek10]. Nevertheless, there are many
${\mathcal B}$
-free Toeplitz systems which do not satisfy (Sh) anyway, see the example discussed at the end of this introduction, so that there is need for techniques relying neither on low complexity nor on (Sh).
We mention briefly that Mentzen [Reference Mentzen22] proves the triviality of the automorphism group for any Erdős
${\mathcal B}$
-free subshift, that is, when
${\mathcal B}$
is infinite, pairwise coprime and
$\sum _{b\in {\mathcal B}}({1}/b)<\infty $
. This is extended to taut
${\mathcal B}$
containing an infinite pairwise coprime subset in [Reference Keller19, Reference Keller, Lemańczyk, Richard and Sell20]. This class of
${\mathcal B}$
-free subshifts is kind of opposite to the
${\mathcal B}$
-free Toeplitz shifts.
1.5 This paper’s contributions to
${\mathcal B}$
-free Toeplitz shifts
In §3, the results from §2 are applied to
${\mathcal B}$
-free examples. This is possible because Theorem 3.17 provides an arithmetic characterization of the sets of essential holes in terms of sets of multiples derived explicitly from the set
${\mathcal B}$
. Then we can use the general results to produce examples of minimal
${\mathcal B}$
-free systems, including not only the case
${\mathcal B}={\mathcal B}_1$
treated in [Reference Dymek10] but also many systems violating the separated holes condition (Sh), which have trivial centralizers, see §3.3. (The reader will notice that a very broad class of examples can be treated along the same lines.) The fact that the general results fail to guarantee triviality of the centralizer for some examples, to which even the basic Theorem 2.8 applies, is not a shortcoming of our approach. This is illustrated in §3.4, where we consider simple variants of
${\mathcal B}={\mathcal B}_1$
, still satisfying condition (Sh), but having non-trivial centralizers—just as big as Theorem 2.8 allows them to be. This provides a negative answer to [Reference Ferenczi, Kułaga Przymus, Lemańczyk, Ferenczi, Kułaga-Przymus and Lemańczyk13, Question 11]. Indeed, a slight generalization of this construction provides examples for which the centralizer contains elements of arbitrarily large finite order, see Remark 3.24. It should be noticed that our examples have superpolynomial complexity, see Proposition 3.30, so that the complexity based results from the literature discussed above do not apply.
1.6 The formal setting
We recall some notation and results from [Reference Baake, Jäger and Lenz1], where a cut-and-project scheme is associated with a Toeplitz sequence
$\eta \in \{0,1\}^{\mathbb Z}$
.
-
(i)
$p_1\mid p_2\mid p_3 \ldots \to \infty $
is a period structure of
$\eta $
. -
(ii)
${\mathcal P}_n^i:=\{k\in {\mathbb Z}: \eta _{|k+p_n{\mathbb Z}}=i\}$
denotes
$p_n$
-periodic positions of
$i=0,1$
on
$\eta $
and
${\mathcal H}_n:={\mathbb Z}\setminus ({\mathcal P}_n^0\cup {\mathcal P}_n^1)$
denotes the set of holes at level n. -
(iii)
$G:=\varprojlim {\mathbb Z}/p_n{\mathbb Z}$
and
$\Delta \colon {\mathbb Z} \rightarrow G$
, where
$\Delta (n) = (n+ p_1{\mathbb Z},n+ p_2{\mathbb Z},\ldots )$
denotes the diagonal embedding. -
(iv)
$T\colon G \rightarrow G$
denotes the rotation by
$\Delta (1)$
, that is,
$(Tg)_n = g_n + 1+p_n{\mathbb Z}$
for all
$n\in {\mathbb N}$
. -
(v) The topology on G is generated by the (open and closed) cylinder sets
$$ \begin{align*}U_n(h) := \{g\in G : g_n = h_n\},\quad h\in G.\end{align*} $$
-
(vi)
$V^i:=\bigcup _{n\in {\mathbb N}}\bigcup _{k\in {\mathcal P}_n^i\cap [0,p_n)}U_n(\Delta (k))$
for
$i=0,1$
. -
(vii) The window
$W:=\overline {V^1}=G\setminus V^0$
is topologically regular, i.e.
$\overline {\operatorname {int}(W)}=W$
. -
(viii)
$\phi \colon G\rightarrow \{0,1\}^{\mathbb Z}$
is the coding function:
$(\phi (g))_n = \mathbf {1}_W(g+\Delta (n))$
. Observe that
$\phi (\Delta (0))=\eta $
.
1.7 More background material and an outline of the paper
The odometer
$(G,T)$
is the MEF of
$(X_{\eta },\sigma )$
, see [Reference Williams25]. Let
$F\colon X_\eta \to X_\eta $
be an automorphism commuting with
$\sigma $
, and denote by
$\pi :(X_\eta ,\sigma )\to (G,T)$
that factor map onto the MEF which is uniquely determined by
$\pi (\eta )=\Delta (0)$
. Then there is
$y_F\in G$
such that the rotation
$f:y\mapsto y+y_F$
on G represents F in the sense that
$\pi \circ F=f\circ \pi $
(see e.g. [Reference Donoso, Durand, Maass and Petite6, Lemma 2.4]). Observe that
$y_F=\pi (F(\eta ))$
.
Denote by
$C_\phi $
the set of continuity points of
$\phi :G\to \{0,1\}^{\mathbb Z}$
. Then,
$|\pi ^{-1}\{\pi (x)\}|=1$
if and only if
$\pi (x)\in C_\phi $
and, in this case,
$x=\phi (\pi (x))$
. This is folklore knowledge, but the reader may consult [Reference Keller and Richard21, Remark 4.2] and also [Reference Downarowicz, Kolyada, Manin and Ward8, §§5–7] for a related and more general perspective on this point.
Since F is a bijection respecting the fibre structure
$X_\eta =\bigcup _{h\in G}\pi ^{-1}\{h\}$
, we have
$|\pi ^{-1}\{\pi (F(x))\}|=|\pi ^{-1}\{\pi (x)\}|$
for all
$x\in X_\eta $
. In particular,
$f(C_\phi )=C_\phi $
, that is,
$C_\phi +y_F=C_\phi $
, see also [Reference Downarowicz, Kolyada, Manin and Ward8, Lemma 4.2]. As the set of discontinuities of the indicator function
$\mathbf {1}_W$
is precisely the boundary
$\partial W$
, a moment’s reflection shows that
$X_\eta \setminus C_\phi =\bigcup _{k\in {\mathbb Z}}\partial W+\Delta (k)$
, so
$$ \begin{align} \partial W+y_F\subseteq \bigcup_{k\in{\mathbb Z}}\partial W+\Delta(k). \end{align} $$
We will use only this property of
$y_F$
to investigate the nature of possible automorphisms F. It is quite natural to expect that this will be much facilitated if the union on the right-hand side of equation (1) is disjoint. Indeed, for general Toeplitz sequences, Bułatek and Kwiatkowski [Reference Bułatek and Kwiatkowski2] studied the centralizer problem under this assumption, because their condition (Sh) is equivalent to the disjointness condition
$$ \begin{align} &\partial W\cap(\partial W+\Delta(k))=\emptyset\quad \text{for all }k\in{\mathbb Z}\setminus\{0\}. \end{align} $$
Namely, condition (Sh) is satisfied if and only if each T-orbit in G hits the set of discontinuities of
$\phi $
at most once, which is clearly equivalent to condition (D), see also [Reference Downarowicz, Kwiatkowski and Lacroix9, remark after Definition 1].
As in [Reference Bułatek and Kwiatkowski2, Proposition 3], it follows that:
-
(1) each fibre over a point in the MEF contains either exactly one or exactly two points; and
-
(2) there exists
$N\in {\mathbb N}$
such that
$\partial W+y_F\subseteq \bigcup _{|k|\leqslant N}\partial W+\Delta (k)$
.
We will introduce a weaker disjointness condition which also implies condition (2), see Proposition 2.12:
$$ \begin{align} &\text{for all } k\in{\mathbb Z}\setminus\{0\}: \partial W\cap(\partial W-\Delta(k))\nonumber\\ &\quad\text{ is nowhere dense with respect to the subspace topology of }\partial W. \end{align} $$
Moreover, we need a strengthened version of this condition which, in many examples, will help to show that
$\partial W+y_F\subseteq \partial W+\Delta (k_0)$
for a single
$k_0\in {\mathbb Z}$
:
$$ \begin{align} &\text{for all } k\in{\mathbb Z}\setminus\{0\}\ \text{ for all } \beta\in G: \partial W\cap(\partial W-\beta)\cap(\partial W-2\beta-\Delta(k))\nonumber\\ &\quad\text{is nowhere dense with respect to the subspace topology of }\partial W. \end{align} $$
Conditions (D′) and (DD′) and an additional growth restriction on the arithmetic structure, see equation (42), play essential roles for proving that the assumption of our basic Theorem 2.8 is satisfied. To prove conditions (D′) and (DD′), we assume in equation (AS) below that the sets of essential holes have some particular arithmetic structure motivated by the intended applications to
${\mathcal B}$
-free Toeplitz shifts, see Propositions 2.20, 2.28 and 2.30. We note here that the additional growth restriction excludes irregular Toeplitz shifts, see Remark 2.29. In Theorem 3.17, we show that
${\mathcal B}$
-free Toeplitz subshifts indeed satisfy the structural assumption in equation (AS).
A characterization of the triviality of the centralizer is provided in [Reference Bułatek and Kwiatkowski2, Theorem 1]. The example
${\mathcal B}_1=\{2^nc_n:n\in {\mathbb N}\}$
with coprime odd
$c_n>1$
from [Reference Dymek10] satisfies condition (D), but we were unable to evaluate the criterion from [Reference Bułatek and Kwiatkowski2, Theorem 1] for it. Instead, we will show that the example not only satisfies
$\partial W+y_F\subseteq \bigcup _{|k|\leqslant N}\partial W+\Delta (k)$
, but that there exists a single integer
$k_0$
such that
$\partial W+y_F\subseteq \partial W+\Delta (k_0)$
, i.e.
$\partial W+(y_F-\Delta (k_0))\subseteq \partial W$
, see Example 3.20. The same holds for the example
${\mathcal B}_1'={\mathcal B}_1\cup \{c_1^2\}$
and for further generalizations of this kind. We shall see that this is the key to control the centralizer with modest efforts in Corollary 2.21: in the case of
${\mathcal B}_1$
, the centralizer is trivial (see also [Reference Dymek10]), while for
${\mathcal B}_1'$
, our approach only yields that the
$c_1$
th iterate of each centralizer element is trivial. Indeed, we will show for this example that the centralizer has an element of order
$c_1$
, see Proposition 3.23 and Remark 3.24. More generally, we will show that elements of the centralizer can be of arbitrarily large finite order, see Corollary 3.25.
Already, the example
${\mathcal B}_2=\{2^nc_n, 3^nc_n:n\in {\mathbb N}\}$
, with coprime
$c_n>1$
also coprime to
$2$
and
$3$
and satisfying
$\prod _{n\in {\mathbb N}}(1-{1}/{c_n})>\tfrac 12$
, violates condition (Sh) but satisfies conditions (D′) and (DD′), see Example 3.21. In §3.6, we show that the shift determined by
${\mathcal B}_2$
has superpolynomial complexity (the same holds for
${\mathcal B}_1$
), so that the results from [Reference Cyr and Kra4–Reference Donoso, Durand, Maass and Petite6] do not apply. We also deliver examples for which not all holes are essential, see Examples 3.27 and 3.29.
2 The abstract regular Toeplitz case
2.1 A first basic theorem
Let
$\eta \in \{0,1\}^{\mathbb Z}$
be a non-periodic Toeplitz sequence with period structure
$p_1\mid p_2\mid p_3 \ldots \to \infty $
. That means for each
$k\in {\mathbb Z}$
, there exists
$n\ge 1$
such that
$\eta _{|k+p_n{\mathbb Z}}$
is constant, but
$\eta $
is not periodic. For
$i=0,1$
, denote
${\mathcal P}_n^i=\{k\in {\mathbb Z}: \eta _{|k+p_n{\mathbb Z}}=i\}$
(we use this notation, because it is shorter than
$\operatorname {Per}_{p_n}(\eta ,i)$
established in the literature) and
${\mathcal H}_n={\mathbb Z}\setminus ({\mathcal P}_n^0\cup {\mathcal P}_n^1)$
. Here,
${\mathcal H}_n$
is called the set of holes at level n.
The odometer group
$G=\varprojlim {\mathbb Z}/p_n{\mathbb Z}$
with the
${\mathbb Z}$
-action ‘addition of
$1$
’ (
$T:G\to G$
,
$(Tg)_n=g_n+1$
) is the MEF of the subshift
$X_\eta $
, which is the orbit closure of
$\eta $
under the left shift
$\sigma $
on
$\{0,1\}^{\mathbb Z}$
. In symbols,
$\pi :(X_\eta ,\sigma )\to (G,T)$
. Observe that
${\mathcal H}_n\subseteq {\mathcal H}_N$
if
$n>N$
.
In [Reference Baake, Jäger and Lenz1], a cut and project scheme is associated with
$\eta $
by specifying a compact and topologically regular window
$W\subseteq G$
: for
$h\in G$
, let
$U_n(h)=\{g\in G: g_n=h_n\}$
and define
$$ \begin{align*} V^i=\bigcup_{n\in{\mathbb N}}\bigcup_{k\in{\mathcal P}_n^i\cap[0,p_n)}U_n(\Delta(k)) \end{align*} $$
for
$i=0,1$
. From [Reference Baake, Jäger and Lenz1, Theorem 1 and its proof], we see:
-
(1)
$W:=\overline {V^1}=G\setminus V^0$
is topologically regular, i.e.
$\overline {\operatorname {int}(W)}=W$
; -
(2)
$\partial W=G\setminus (V^0\cup V^1)$
; -
(3)
$\Delta (k)\in V^i$
if and only if
$\eta _k=i$
for
$i=0,1$
and all
$k\in {\mathbb Z}$
, in particular
$\eta _k=1$
if and only if
$\Delta (k)\in W$
.
Lemma 2.1.
$h\in \partial W$
if and only if
$h_N\in {\mathcal H}_N$
for all
$N\in {\mathbb N}$
.
Proof. We have
$$ \begin{align*} h\not\in\partial W &\Leftrightarrow h\in\bigcup_{i\in\{0,1\}}V^i \Leftrightarrow h\in\bigcup_{i\in\{0,1\}}\bigcup_{n\in{\mathbb N}}\bigcup_{k\in{\mathcal P}_n^i\cap[0,p_n)}U_n(\Delta(k))\\ &\Leftrightarrow \text{there exists } i\in\{0,1\}\text{ there exists } n\in{\mathbb N}\text{ there exists }\\ &\qquad k\in{\mathcal P}_n^i\cap[0,p_n): h_n=k\\ &\Leftrightarrow \text{there exists } i\in\{0,1\}\text{ there exists } n\in{\mathbb N}: h_n\in{\mathcal P}_n^i\\ &\Leftrightarrow \text{there exists } n\in{\mathbb N}: h_n\not\in {\mathcal H}_n\\[-2.8pc] \end{align*} $$
Definition 2.2. (Essential holes)
The set of essential holes at level N is defined as
$$ \begin{align} {\widetilde{\mathcal H}}_N:=\{k\in{\mathcal H}_N: {\mathcal H}_n\cap(k+p_N{\mathbb Z})\neq\emptyset\text{ for all }n\ge N\}. \end{align} $$
Definition 2.3. The minimal periods of
${\mathcal H}_n$
and
${\widetilde {\mathcal H}}_n$
are denoted by
$\tau _n$
and
$\tilde {\tau }_n$
, respectively.
Remark 2.4.
-
(a)
${\widetilde {\mathcal H}}_n\subseteq {\widetilde {\mathcal H}}_N$
if
$n>N$
. -
(b)
${\mathcal H}_N$
and
${\widetilde {\mathcal H}}_N$
are
$p_N$
-periodic by definition—although this need not be their minimal period. (For
${\widetilde {\mathcal H}}_N$
, just observe that
$k\in {\widetilde {\mathcal H}}_N$
if and only if
$k+p_N\in {\widetilde {\mathcal H}}_N$
.) Hence,
$\tau _N\mid p_N$
and
$\tilde {\tau }_N\mid p_N$
, so that expressions like ‘
$\tau _N\mid h_N$
’ or ‘
$\gcd (\tilde {\tau }_N,h_N)$
’ are well defined when
$h_N\in {\mathbb Z}/p_N{\mathbb Z}$
. (This is consistent with the following general convention: an element z of an abelian group Z is divisible by
$n\in {\mathbb N}$
if
$z\in nZ$
. If Z is a cyclic group and
$z_1,z_2\in Z$
, then
$\gcd (z_1,z_2)$
is a generator of the subgroup of Z generated by
$z_1$
and
$z_2$
. If
$Z={\mathbb Z}$
, then we choose
$\gcd (z_1,z_2)$
to be positive by convention. Finally, if
$n\in {\mathbb Z}$
and
$z+p{\mathbb Z}\in {\mathbb Z}/p{\mathbb Z}$
, then we understand by
$\gcd (n,z+p{\mathbb Z})$
the (positive) generator of the group generated by n and
$z+p{\mathbb Z}$
; this is the greatest common divisor of the numbers
$n,z$
and p in the usual sense.)
Lemma 2.5.
-
(a)
$k\in {\widetilde {\mathcal H}}_N$
if and only if
$U_N(\Delta (k))\cap \partial W\neq \emptyset $
. -
(b)
$h\in \partial W$
if and only if
$h_N\in {\widetilde {\mathcal H}}_N$
for all
$N\in {\mathbb N}$
.
Proof. (a) Let
$h\in U_N(\Delta (k))\cap \partial W$
. Then
$k\in h_N+p_N{\mathbb Z}\subseteq {\mathcal H}_N+p_N{\mathbb Z}={\mathcal H}_N$
by Lemma 2.1, and for all
$n\ge N$
, we have in view of Lemma 2.1:
$h_n\in {\mathcal H}_n\cap (h_N+p_N{\mathbb Z})={\mathcal H}_n\cap (k+p_N{\mathbb Z})$
. For the reverse implication, let
$k\in {\widetilde {\mathcal H}}_N$
. We construct
$h\in \partial W$
with
$h_N=k\,\mod p_N$
: for
$j>N$
, there is
$r_j\in {\mathcal H}_j\cap (k+p_N{\mathbb Z})$
. Let
$h^{(j)}=\Delta (r_j)$
and fix any accumulation point h of the sequence
$(h^{(j)})_j$
. Consider any
$n\ge N$
. For some sufficiently large
$j_n\ge n$
, we have
$h_n=h_n^{(j_n)}=r_{j_n}\,\mod p_n$
. Hence,
$h_N\in r_{j_N}+p_N{\mathbb Z}=k+p_N{\mathbb Z}$
, so that
$h\in U_N(\Delta (k))$
, and
$h_n\in r_{j_n}+p_n{\mathbb Z}\subseteq {\mathcal H}_{j_n}+p_n{\mathbb Z}\subseteq {\mathcal H}_n+p_n{\mathbb Z}={\mathcal H}_n$
, so that
$h\in \partial W$
by Lemma 2.1.
(b) We have
$$ \begin{align*} h\in\partial W &\Leftrightarrow\text{for all } N\in{\mathbb N}:U_N(h)\cap\partial W\neq\emptyset\\&\Leftrightarrow\text{for all } N\in{\mathbb N}:U_N(\Delta(h_N))\cap\partial W\neq\emptyset\\&\Leftrightarrow\text{for all } N\in{\mathbb N}:h_N\in{\widetilde{\mathcal H}}_N\quad\text{by part (a).}\\[-2.9pc] \end{align*} $$
The following simple lemma is basic for our approach.
Lemma 2.6.
-
(a) If
$h+\beta {\mathbb Z}\subseteq \partial W$
for some
$h,\beta \in G$
, then
$h_n+\gcd (\beta _n,\tilde {\tau }_n){\mathbb Z}\subseteq {\widetilde {\mathcal H}}_n$
for all
$n>0$
. -
(b) If
$\partial W+\beta \subseteq \partial W$
for some
$\beta \in G$
, then
$\tilde {\tau }_n\mid \beta _n$
for all
$n>0$
.
Proof. (a)
$h_n+\beta _n{\mathbb Z}\subseteq {\widetilde {\mathcal H}}_n$
for all n by Lemma 2.5(b). As
${\widetilde {\mathcal H}}_n$
is
$\tilde {\tau }_n$
-periodic, this implies
$h_n+\gcd (\beta _n,\tilde {\tau }_n){\mathbb Z}\subseteq {\widetilde {\mathcal H}}_n$
.
(b) Let
$j\in {\widetilde {\mathcal H}}_n$
. Then there is some
$h\in U_n(\Delta (j))\cap \partial W$
by Lemma 2.5(a). Now,
$h+\beta {\mathbb Z}\subseteq \partial W$
by assumption. As
$h_n=j$
, this implies, in view of part (a) of the lemma,
$j+\gcd (\beta _n,\tilde {\tau }_n){\mathbb Z}\subseteq {\widetilde {\mathcal H}}_n$
. As this holds for each
$j\in {\widetilde {\mathcal H}}_n$
, we have
${\widetilde {\mathcal H}}_n+\gcd (\beta _n,\tilde {\tau }_n){\mathbb Z}={\widetilde {\mathcal H}}_n$
, and as
$\tilde {\tau }_n$
is the minimal period of
${\widetilde {\mathcal H}}_n$
, we conclude that
$\tilde {\tau }_n\mid \beta _n$
.
Remark 2.7. If
$h+\beta {\mathbb Z}\subseteq \partial W$
, Lemma 2.6 implies
$$ \begin{align*} \boldsymbol{\delta}({\mathcal H}_n)\ge \boldsymbol{\delta}({\widetilde{\mathcal H}}_n)\ge \frac{1}{\gcd(\beta_n,\tilde{\tau}_n)}. \end{align*} $$
Therefore, the last lemma seems to be useful only for regular Toeplitz shifts, because for irregular ones,
$\inf _n\boldsymbol {\delta }({\mathcal H}_n)>0$
so that no useful lower bound on
$\gcd (\beta _n,\tilde {\tau }_n)$
can be expected.
Each automorphism F of
$(X_\eta ,\sigma )$
determines an element
$y_F\in G$
such that
$\pi (F(x))=\pi (x)+y_F$
for all
$x\in X_\eta $
. Observe that
$$ \begin{align} \partial W+y_F\subseteq\bigcup_{k\in{\mathbb Z}}(\partial W+\Delta(k)), \end{align} $$
because F leaves the set of non-one-point fibres over the MEF invariant. We first focus on a stronger property than equation (3).
Theorem 2.8. Recall that
$\tilde {\tau }_n$
denotes the minimal period of
${\widetilde {\mathcal H}}_n$
.
-
(a) If an automorphism F of
$(X_\eta ,\sigma )$
satisfies
$\partial W+y_F\subseteq \partial W+\Delta (k)$
for some
$k=k_F\in {\mathbb Z}$
, then
$\tilde {\tau }_n\mid (y_F)_n-k_F$
. In particular, if infinitely many
${\widetilde {\mathcal H}}_n$
have minimal period
$p_n$
, then
$y_F=\Delta (k_F)$
. -
(b) Suppose that for each automorphism
$F\in \operatorname {Aut}_\sigma (X_\eta )$
, there exists a unique
$k_F\in {\mathbb Z}$
such that (4)
$$ \begin{align} \partial W+y_F\subseteq \partial W+\Delta(k_F). \end{align} $$
If
$M:=\liminf _{n\to \infty }p_n/\tilde {\tau }_n<\infty $
, then where
$$ \begin{align*} \operatorname{Aut}_\sigma(X_\eta)=\langle \sigma\rangle\oplus\operatorname{Tor}, \end{align*} $$
$\operatorname {Tor}$
denotes the torsion group of
$\operatorname {Aut}_\sigma (X_\eta )$
. Moreover,
$\operatorname {Tor}$
is a cyclic group (possibly trivial), whose order divides M. In particular, if infinitely many
${\widetilde {\mathcal H}}_n$
have minimal period
$p_n$
, then the centralizer of
$(X_\eta ,\sigma )$
is trivial.
Proof. (a) The first claim follows from Lemma 2.6(b), the second one is just a special case of this.
(b) In each residue class of
$\operatorname {Aut}_\sigma (X_\eta )/\langle \sigma \rangle $
, there is exactly one element F for which the associated integer
$k_F$
satisfying equation (4) equals
$0$
. These elements F form a subgroup J of
$\operatorname {Aut}_\sigma (X_\eta )$
. Suppose for a contradiction that there are
$M+1$
different automorphisms
$F_1,\ldots ,F_{M+1}\in J$
. In view of Lemma 2.6(b), they all satisfy
$$ \begin{align} \tilde{\tau}_n\mid (y_{F_{i}})_{S_n} \quad\text{for all }n\in{\mathbb N}. \end{align} $$
Hence, there exists arbitrarily large
$n\in {\mathbb N}$
such that
$M=p_n/\tilde {\tau }_n$
and
$$ \begin{align*} (y_{F_i})_{S_n}/\tilde{\tau}_n\in\{0,\ldots,M-1\} \quad\text{for all }i=1,\ldots,M+1. \end{align*} $$
It follows that there exist two different
$i,j\in \{1,\ldots ,M+1\}$
for which
$(y_{F_i})_{S_n}=(y_{F_j})_{S_n}$
for infinitely many n, which is only possible if
$y_{F_i}=y_{F_j}$
. In view of [Reference Donoso, Durand, Maass and Petite6, Lemma 2.4], this implies
$F_i=F_j\,\mod \langle \sigma \rangle $
, so that
$i=j$
, which is a contradiction. Hence, J is a finite group of order
$m\leqslant M$
, say
$J=\{F_1,\ldots ,F_m\}$
. In particular,
$J\subseteq \operatorname {Tor}$
. However, if
$F\in \operatorname {Tor}$
, then
$ry_F=\Delta (0)\in G$
for some positive integer r. Hence, with the integer
$k_F$
satisfying equation (4), we have
$$ \begin{align*} \partial W+y_{Id}-\Delta(rk_F)=\partial W-\Delta(rk_F)=\partial W+ry_F-\Delta(rk_F)\subseteq \partial W, \end{align*} $$
but
$k_{Id}=0$
, so equation (4) implies
$k_F=0$
. It follows that
$F\in J$
, and we proved that
$J=\operatorname {Tor}$
. Then [Reference Donoso, Durand, Maass and Petite7, Theorem 3.2(2)] implies that
$\operatorname {Tor}$
is cyclic, and
$\operatorname {Aut}_\sigma (X_\eta )=J\oplus \langle \sigma \rangle =\operatorname {Tor}\oplus \ \langle \sigma \rangle $
.
It remains to determine the order m of
$\operatorname {Tor}$
: fix some
$F\in \operatorname {Tor}$
. In view of equation (5),
$$ \begin{align*} p_n=M\cdot\tilde{\tau}_n\mid M\cdot(y_F)_{S_n} \equiv (y_{F^M})_{S_n}\,\mod\! p_n \end{align*} $$
for all
$n\in {\mathbb N}$
. Hence,
$F^M=\operatorname {id}_{X_\eta }$
, so that the order of F is a divisor of M.
To apply this theorem, we need to verify the assumption in equation (4) and to determine the periods
$\tilde {\tau }_n$
. So we focus next on finding sufficient conditions that imply
$\partial W+ (y_F-\Delta (k)){\mathbb Z}\subseteq \partial W$
.
2.2 The separated holes conditions and its variants
The separated holes condition (Sh) was introduced in [Reference Bułatek and Kwiatkowski2]. It requires
$$ \begin{align} &\text{ for all } k\in{\mathbb Z}\setminus\{0\},\quad \text{there exists } N\in{\mathbb N}\quad \text{for all } n\ge N: {\mathcal H}_n\cap({\mathcal H}_n-k)=\emptyset. \end{align} $$
As mentioned in the introduction, it is equivalent to the disjointness condition
$$ \begin{align} &\text{ for all }\ k\in{\mathbb Z}\setminus\{0\}: \partial W\cap (\partial W-\Delta(k))=\emptyset. \end{align} $$
Indeed, it is even equivalent to the separated essential holes condition
$$ \begin{align} \text{ for all } k\in{\mathbb Z}\setminus\{0\},\quad \text{there exists } N\in{\mathbb N}\quad \text{for all } n\ge N: {\widetilde{\mathcal H}}_n\cap({\widetilde{\mathcal H}}_n-k)=\emptyset. \end{align} $$
As we do not make use of this equivalence, we leave it as an exercise.
The following variants of the separated essential holes condition, which allow to study Toeplitz subshifts that violate condition (Sh), will play important roles; however, so we provide proofs for the corresponding equivalences:
-
(i) weak disjointness condition (D′), equivalent to weak separated essential holes condition (Seh′):
(D′)
$$ \begin{align} &\text{for all } k\in{\mathbb Z}\setminus\{0\}:\partial W\cap(\partial W-\Delta(k))\nonumber \\ &\quad\text{ is nowhere dense with respect to~the subspace topology of }\partial W. \end{align} $$
(Seh′)
$$ \begin{align} &\text{for all } k\in{\mathbb Z}\setminus\{0\}:\text{there is no arithmetic progression }r+p_N{\mathbb Z}\text{ such that} \nonumber \\ &\quad\text{for all } n\ge N:\emptyset\neq(r+p_N{\mathbb Z})\cap{\widetilde{\mathcal H}}_n\subseteq{\widetilde{\mathcal H}}_n-k; \end{align} $$
-
(ii) weak double disjointness condition (DD′), equivalent to weak double separated essential holes condition (DSeh′):
(DD′)
$$ \begin{align} &\text{for all } k\in{\mathbb Z}\setminus\{0\}\quad \text{for all } \beta\in G: \partial W\cap(\partial W-\beta)\cap(\partial W-2\beta-\Delta(k))\nonumber\\ &\quad\text{is nowhere dense with respect to the subspace topology of }\partial W, \end{align} $$
(DSeh′)
$$ \begin{align} &\text{for all } k\in{\mathbb Z}\setminus\{0\}\ \text{ for all } \beta\in G:\text{there is no arithmetic progression }r+p_N{\mathbb Z}\nonumber\\ &\quad\text{such that for all } n\ge N:\emptyset\neq(r+p_N{\mathbb Z})\cap{\widetilde{\mathcal H}}_n\subseteq ({\widetilde{\mathcal H}}_n-\beta_n)\cap({\widetilde{\mathcal H}}_n-2\beta_n-k). \end{align} $$
Observe that for
$\beta =0$
conditions (DD′) and (DSeh′) reduce to conditions (D′) and (Seh′), respectively. Moreover, conditions (D) and (Sh) clearly imply conditions (D′) and (Seh′), respectively.
In the following, we will assume condition (DD′)—indeed, for some results, only the weaker condition (D′) is needed. In the
${\mathcal B}$
-free setting, condition (DD′) will be verified under suitable assumptions in Proposition 2.30.
Both equivalences above, namely
$\lnot $
condition (D′)
$\Leftrightarrow \lnot $
condition (Seh′) and
$\lnot $
condition (DD′)
$\Leftrightarrow \lnot $
condition (DSeh′), follow immediately from the next lemma.
Lemma 2.9. Let
$k\in {\mathbb Z}\setminus \{0\}$
,
$\beta \in G$
,
$N>0$
and
$r\in {\mathbb Z}$
. Then,
$$ \begin{align} \emptyset\neq U_N(\Delta(r))\cap\partial W\subseteq(\partial W-\beta)\cap(\partial W-2\beta-\Delta(k)) \end{align} $$
if and only if
$$ \begin{align} \text{ for all } n\ge N:\emptyset\neq (r+p_N{\mathbb Z})\cap{\widetilde{\mathcal H}}_n\subseteq({\widetilde{\mathcal H}}_n-\beta_n)\cap({\widetilde{\mathcal H}}_n-2\beta_n-k). \end{align} $$
Proof. Suppose that equation (6) holds. Then, for all
$n\ge N$
, there is
$r_n\in r+p_N{\mathbb Z}$
such that
$U_n(\Delta (r_n))\cap \partial W\neq \emptyset $
, whence
$r_n\in (r+p_N{\mathbb Z})\cap {\widetilde {\mathcal H}}_n$
in view of Lemma 2.5(a).
Now consider any
$r_n\in (r+p_N{\mathbb Z})\cap {\widetilde {\mathcal H}}_n$
. By the inclusion in equation (6),
$$ \begin{align*} U_n(\Delta(r_n))&\cap\partial W\subseteq U_N(\Delta(r_n))\cap\partial W\\ &=U_N(\Delta(r))\cap\partial W\subseteq(\partial W-\beta)\cap(\partial W-2\beta-\Delta(k)), \end{align*} $$
so that, by Lemma 2.5(a) again,
$r_n+\beta _n\in {\widetilde {\mathcal H}}_n$
and
$r_n+2\beta _n+k\in {\widetilde {\mathcal H}}_n$
, which proves equation (7).
Conversely, suppose that equation (7) holds. For each
$n\ge N$
, there is some
$r_n\in (r+p_N{\mathbb Z})\cap {\widetilde {\mathcal H}}_n$
, and we find a subsequence
$\Delta (r_{n_i})$
that converges to some
$h\in G$
. Hence, for each
$m>0$
, there is
$n_i\ge m$
such that
$h_m\in r_{n_i}+p_m{\mathbb Z}\subseteq {\widetilde {\mathcal H}}_{n_i}+p_m{\mathbb Z}\subseteq {\widetilde {\mathcal H}}_m+p_m{\mathbb Z}={\widetilde {\mathcal H}}_m$
, so that
$h\in \partial W$
in view of Lemma 2.5(b). As
$h_N\in r_{n_i}+p_N{\mathbb Z}=r+p_N{\mathbb Z}$
for some
$n_i$
, this shows that
$h\in U_N(\Delta (r))\cap \partial W$
.
Now consider any
$h\in U_N(\Delta (r))\cap \partial W$
. Then, for all
$n\ge N$
,
$U_n(h)\cap \partial W\neq \emptyset $
, so that
$h_n\in (r+p_N{\mathbb Z})\cap {\widetilde {\mathcal H}}_n$
, where we used Lemma 2.5(a) once more. The inclusion in equation (7) then implies
$h_n\in ({\widetilde {\mathcal H}}_n-\beta _n)\cap ({\widetilde {\mathcal H}}_n-2\beta _n-k)$
for all
$n\ge N$
. Now Lemma 2.5(b) shows that
$h+\beta \in \partial W$
and
$h+2\beta +\Delta (k)\in \partial W$
, which proves equation (6).
Remark 2.10. Suppose that the condition (*) from [Reference Bułatek and Kwiatkowski3] holds, that is, for any
$n\in {\mathbb N}$
and
$s\in {\mathbb Z}$
,
$$ \begin{align} &[sp_n,(s+1)p_n)\cap{\mathcal H}_n\subseteq{\mathcal H}_{n+1}\text{ or }[sp_n,(s+1)p_n)\cap{\mathcal H}_{n+1}=\emptyset, \end{align} $$
and recall that it implies triviality of the centralizer [Reference Bułatek and Kwiatkowski3, Theorem 1]. Here we show that it implies
${\widetilde {\mathcal H}}_n={\mathcal H}_n$
for all n, but mostly precludes property (Seh′): note first that
$$ \begin{align} \text{ for all } n\in{\mathbb N}\quad \text{for all } r\in{\mathcal H}_n,\quad\text{there exists } s\in{\mathbb Z}:r+sp_n\in{\mathcal H}_{n+1}. \end{align} $$
Indeed, otherwise there are
$n\in {\mathbb N}$
and
$r\in {\mathcal H}_n$
such that
$r+sp_n\in {\mathcal H}_n\setminus {\mathcal H}_{n+1}$
for all
$s\in {\mathbb Z}$
. However, then
${\mathcal H}_{n+1}=\emptyset $
in view of condition (*), which is excluded because we study only non-periodic Toeplitz sequences. A straightforward inductive application of condition (8) shows that for all
$n\in {\mathbb N}$
,
$r\in {\mathcal H}_n$
and
$m> n$
, there are integers
$s_n,\ldots ,s_{m-1}$
such that
$r+s_np_n+\cdots +s_{m-1}p_{m-1}\in (r+p_n{\mathbb Z})\cap {\mathcal H}_m$
. Hence,
${\widetilde {\mathcal H}}_n={\mathcal H}_n$
for all
$n\in {\mathbb N}$
. We claim
$$ \begin{align} (r+p_N{\mathbb Z})\cap{\widetilde{\mathcal H}}_n\subseteq{\widetilde{\mathcal H}}_n-(r'-r)\quad\text{for any }r,r'\in{\mathcal H}_N\cap[0,p_N). \end{align} $$
Indeed, let
$r,r'\in {\mathcal H}_N\cap [0,p_N)$
and
$s\in {\mathbb Z}$
. Suppose that
$r+sp_N\in {\mathcal H}_n$
for some
$n>N$
. Since for any
$m\geq N$
we have
$r+sp_N\in [s_m'p_m,(s_m'+1)p_m)$
, where
$s_m'=[{sp_N}/{p_m}]$
, and
${\mathcal H}_n\subseteq {\mathcal H}_{n-1}\subseteq \cdots \subseteq {\mathcal H}_N$
, by condition (*), we obtain
$[s_m'p_m,(s_m'+1)p_m)\cap {\mathcal H}_m\subseteq {\mathcal H}_{m+1}$
for any
$N\leq m< n$
. In particular, since
$[sp_N,(s+1)p_N)\subseteq [s_m'p_m,(s_m'+1)p_m)$
, we have
$[sp_N,(s+1)p_N)\cap {\mathcal H}_m\subseteq {\mathcal H}_{m+1}$
for any
$N\leq m<n$
. Hence,
$$ \begin{align*}[sp_N,(s+1)p_N)\cap{\mathcal H}_N=[sp_N,(s+1)p_N)\cap{\mathcal H}_{n-1}\subseteq{\mathcal H}_{n}.\end{align*} $$
Of course,
$r'+sp_N\in [sp_N,(s+1)p_N)\cap {\mathcal H}_N$
. So
$r'+sp_N\in {\mathcal H}_n={\widetilde {\mathcal H}}_n$
. The same arguments as above, with roles of r and
$r'$
interchanged, show that
$$ \begin{align*}r+sp_N\in{\widetilde{\mathcal H}}_n\Leftrightarrow r'+sp_N\in{\widetilde{\mathcal H}}_n \Leftrightarrow r+sp_N\in{\widetilde{\mathcal H}}_n-(r'-r).\end{align*} $$
So equation (9) follows. Hence, if
$[0,p_N)$
contains at least two holes at level N, then condition (Seh′) does not hold. If, however,
${\mathcal H}_n\cap [0,p_n)$
is a singleton for any
$n\geq N$
, then the distance between consecutive holes at level n is
$p_n$
, so even condition (Sh) holds.
Remark 2.11. Given
$k\in {\mathcal H}_N\setminus {\widetilde {\mathcal H}}_N$
, let
$n_k\ge N$
be minimal such that
$(k+p_N{\mathbb Z})\cap {\mathcal H}_{n_k}=\emptyset $
. Clearly,
$n_k$
depends only on the residue of k modulo
$p_N$
, so the
$n_k$
are bounded, say, by
$m_N$
. Then, for
$k\in {\mathcal H}_N$
,
$k\in {\widetilde {\mathcal H}}_N$
if and only if
$(k+p_N{\mathbb Z})\cap {\mathcal H}_{m_N}\neq \emptyset $
. More generally, for every
$n\ge m_N$
:
$k\in {\widetilde {\mathcal H}}_N$
if and only if
$(k+p_N{\mathbb Z})\cap {\mathcal H}_{n}\neq \emptyset $
. Hence, for every
$n\ge m_N$
,
${\widetilde {\mathcal H}}_N={\mathcal H}_{n}+p_N{\mathbb Z}$
. It follows that the minimal period
$\tilde {\tau }_N$
of
${\widetilde {\mathcal H}}_N$
divides
$\gcd (\tau _n,p_N)$
for
$n\ge m_N$
.
2.3 Consequences of the weak disjointness condition (D′)
Consider any automorphism F of
$(X_\eta ,\sigma )$
. Recall from the introduction that
$\pi \circ F=f\circ \pi $
, where
$\pi :(X_\eta ,\sigma )\to (G,T)$
is the MEF-map and
$f:G\to G, y\mapsto y+y_F$
for some
$y_F\in G$
, and that
$\partial W+y_F\subseteq \bigcup _{k\in {\mathbb Z}}\partial W+\Delta (k)$
. Denote
$$ \begin{align} V_k=\partial W\cap(\partial W+\Delta(k)-y_F)\quad(k\in{\mathbb Z}) \end{align} $$
and
$$ \begin{align} K=\{k\in{\mathbb Z}: \operatorname{int}_{\partial W}(V_k)\neq\emptyset\}. \end{align} $$
Proposition 2.12. Assume that the weak disjointness condition (D′) holds. Let the automorphism F of
$(X_\eta ,\sigma )$
be described by a block code
$\{0,1\}^{[-m:m]}\to \{0,1\}$
. Then,
$$ \begin{align} \partial W+y_F\subseteq\bigcup_{k=-m}^m\partial W+\Delta(k). \end{align} $$
Proof. Recall from equation (3) that
$\partial W+y_F \subseteq \bigcup _{k\in {\mathbb Z}}\partial W+\Delta (k)$
.
Let
$y\in G$
and recall that
$\pi \colon {X_\eta \to G}$
denotes the factor map onto the MEF. At the end of the proof, we show
$$ \begin{align} |\pi^{-1}\{y\}|=2 \Leftrightarrow \text{there exists } j\in{\mathbb Z}:y+\Delta(j)\in R:=\partial W\setminus\bigcup_{k\in{\mathbb Z}\setminus\{0\}}\partial W+\Delta(k). \end{align} $$
As
$F\colon X_\eta \to X_\eta $
is a bijection that maps
$\pi $
-fibres to
$\pi $
-fibres and as
$\pi \circ F=f\circ \pi $
, it follows that when
$\pi ^{-1}\{y\}=\{x_1,x_2\}$
with
$x_1\neq x_2$
, then
$\pi ^{-1}\{f(y)\}=\{x_1'=F(x_1),x_2'=F(x_2)\}$
, and there are exactly one index
$j\in {\mathbb Z}$
such that
$y+\Delta (j)\in \partial W$
and
$(x_1)_j\neq (x_2)_j$
, and exactly one index
$k\in {\mathbb Z}$
such that
$f(y)+\Delta (k)\in \partial W$
and
$(x_1')_{k}\neq (x_2')_{k}$
. As F is described by a block code
$\{0,1\}^{[-m:m]}\to \{0,1\}$
, it follows that
$|j-k|\leqslant m$
. (This argument is taken from the proof of [Reference Bułatek and Kwiatkowski2, Corollary 1].)
Consider any
$y\in R$
. Then the index j in equation (13) equals
$0$
and
$y+y_F\in \partial W-\Delta (k)$
for some k with
$|k|\leqslant m$
. In other words:
$f(R)$
is contained in the closed set
$\bigcup _{k=-m}^mT^k(\partial W)$
. Because of condition (D′), the set R defined in equation (13) is residual with respect to the subspace topology of
$\partial W$
. Hence,
$$ \begin{align*} \partial W+y_F=f(\partial W)=f(\overline{R})\subseteq\overline{f(R)}\subseteq \bigcup_{k=-m}^mT^k(\partial W) =\bigcup_{k=-m}^m\partial W+\Delta(k). \end{align*} $$
It remains to prove equation (13). Recall first that
$|\pi ^{-1}\{y\}|=1$
if and only if
$y\in C_\phi $
, so that
$|\pi ^{-1}\{y\}|>1$
if and only if
$y\in \bigcup _{k\in {\mathbb Z}}\partial W+\Delta (k)$
. So all we must show is that (A)
$|\pi ^{-1}\{y\}|>2$
if and only if (B)
$y+\Delta (j)\in \partial W\cap (\partial W+\Delta (k))$
for some
$j\in {\mathbb Z}$
and
$k\in {\mathbb Z}\setminus \{0\}$
.
Suppose first that part (A) holds, i.e. that there are at least three different points in
$\pi ^{-1}\{y\}$
. As points in the same
$\pi $
-fibre can differ only at positions k where
$y+\Delta (k)\in \partial W$
, there must be at least two such positions, and that is part (B).
Conversely, if part (B) holds and if
$x=\phi (y)$
, then
$x_j=x_{j-k}=1$
.
-
(i) As
$\partial W\cup (\partial W+\Delta (k))$
is nowhere dense in G, there are arbitrarily small perturbations
$y'$
of y such that
$y'+\Delta (j)\not \in \partial W\cup (\partial W+\Delta (k))$
, resulting in points
$x'=\phi (y')$
with
$x^{\prime }_j=x^{\prime }_{j-k}=0$
. -
(ii) As
$\partial W\cap (\partial W+\Delta (k))$
is nowhere dense in
$\partial W$
with respect to the subspace topology on
$\partial W$
(because of condition (D′)), there are arbitrarily small perturbations
$y'$
of y such that
$y'+\Delta (j)\in \partial W\setminus (\partial W+\Delta (k))$
, resulting in
$x'=\phi (y')$
with
$x^{\prime }_j=1$
and
$x^{\prime }_{j-k}=0$
.
Hence,
$|\pi ^{-1}\{y\}|\geqslant 3$
, and that is part (A).
Proposition 2.13. We have
$K\neq \emptyset $
, and there is a countable collection
$U_{{N_j}}(\Delta (r_j))$
,
${j\in {\mathbb N}}$
, of cylinder sets in G with the following properties: for each
$j\in {\mathbb N}$
, there exists some
$k_j\in {\mathbb Z}$
such that
$$ \begin{align} \emptyset\neq(U_{{N_j}}(\Delta(r_j))\cap\partial W)+(y_F-\Delta(k_j))\subseteq\partial W, \end{align} $$
and
$$ \begin{align} \bigcup_{j\in{\mathbb N}}U_{{N_j}}(\Delta(r_j))\cap\partial W\quad \text{is dense in }\partial W. \end{align} $$
Proof. We start from the observation of equation (3), namely
$\partial W+y_F\subseteq \bigcup _{k\in {\mathbb Z}}\partial W+\Delta (k)$
. This implies
$$ \begin{align*} \partial W= \bigcup_{k\in{\mathbb Z}}\partial W\cap(\partial W+\Delta(k)-y_F) =\bigcup_{k\in {\mathbb Z}}V_k. \end{align*} $$
Hence,
$M:=\bigcup _{k\in {\mathbb Z}}V_k\setminus \operatorname {int}_{\partial W}(V_k)$
is a meagre subset of the compact space
$\partial W$
and
$\partial W=M\cup \bigcup _{k\in K}\operatorname {int}_{\partial W}(V_k)$
. Now Baire’s category theorem implies that
$K\neq \emptyset $
. As
$\partial W$
is separable, there is a countable collection
$U_{{N_j}}(\Delta (r_j))$
,
${j\in {\mathbb N}}$
, of cylinder sets in G, for each of which there exists
$k_j\in K$
such that
$\emptyset \neq \partial W\cap U_{{N_j}}(\Delta (r_j))\subseteq \operatorname {int}_{\partial W}(V_{k_j})$
and such that
$$ \begin{align*} \partial W=M\cup\bigcup_{j\in{\mathbb N}}(\partial W\cap U_{{N_j}}(\Delta(r_j))). \end{align*} $$
As M is meagre in
$\partial W$
, these cylinder sets satisfy equation (15), and as
$$ \begin{align*} (U_{{N_j}}(\Delta(r_j))\cap\partial W) \subseteq \operatorname{int}_{\partial W}(V_{k_j}) \subseteq \partial W+\Delta(k_j)-y_F, \end{align*} $$
also equation (14) holds.
Corollary 2.14. Assume that the weak disjointness condition (D′) holds. Let the automorphism F of
$(X_\eta ,\sigma )$
be described by a block code
$\{0,1\}^{[-m:m]}\to \{0,1\}$
. Then the set K is contained in
$[-m,m]$
,
${\operatorname {int}_{\partial W}(V_{k_i})\cap \operatorname {int}_{\partial W}(V_{k_j})=\emptyset }$
for any different
$k_i,k_j\in K$
, and
$\partial W=\bigcup _{k\in K}V_k'$
, where
$V_k':=\overline {\operatorname {int}_{\partial W}(V_k)}$
.
Proof. Let
$K':=K\cap [-m,m]$
, m as in Proposition 2.12. Because of that proposition and Proposition 2.13,
$\partial W=\bigcup _{k\in K'}\overline {\operatorname {int}_{\partial W}(V_k)}$
. Suppose there are
$k_i,k_j\in K$
such that
$\operatorname {int}_{\partial W}(V_{k_i})\cap \operatorname {int}_{\partial W}(V_{k_j})\neq \emptyset $
. Then there is some cylinder set
$U_{N}(\Delta (r))\cap \partial W$
contained in this intersection. Let
$\tilde {U}:=(U_{N}(\Delta (r))\cap \partial W)+y_F-\Delta (k_i)$
. Then,
$\tilde {U}\subseteq \partial W$
and
$\tilde {U}+\Delta (k_i-k_j)\subseteq \partial W$
, so that
$\tilde {U}\subseteq \partial W\cap (\partial W+\Delta (k_j-k_i))$
. As
$\tilde {U}$
is non-empty and open in the relative topology on
$\partial W$
, the weak disjointness assumption implies
$k_i=k_j$
. This also proves that
$K=K'\subseteq [-m,m]$
.
For later use, we note a further consequence of Proposition 2.13.
Corollary 2.15. Assume that
$(\operatorname {int}_{\partial W}(V_{k_i})+(y_F-\Delta (k_i))\cap \operatorname {int}_{\partial W}(V_{k_j})\neq \emptyset $
for some
$k_i,k_j\in K$
. Then there exist
$N\in {\mathbb N}$
(which can be chosen arbitrarily large) and
$r\in {\mathbb Z}$
such that
$$ \begin{align} \emptyset\neq(U_{{N}}(\Delta(r))\cap\partial W)+(y_F-\Delta(k_i))\subseteq&\ \partial W\quad\text{and} \nonumber \\ \emptyset\neq(U_{{N}}(\Delta(r))\cap\partial W)+(y_F-\Delta(k_i))+(y_F-\Delta(k_j))\subseteq&\ \partial W. \end{align} $$
Proof. Fix some cylinder set
$U_{{N'}}(\Delta (r'))$
such that
$$ \begin{align*} \emptyset\neq U_{{N'}}(\Delta(r'))\cap\partial W\subseteq \operatorname{int}_{\partial W}(V_{k_i})\cap (\operatorname{int}_{\partial W}(V_{k_j})-(y_F-\Delta(k_i))). \end{align*} $$
In view of Proposition 2.13, there are
$N_i',N_j'\in {\mathbb N}$
and
$r_i',r_j'\in {\mathbb Z}$
such that
$$ \begin{align} U_{{N'}}(\Delta(r'))\cap U_{{N_i'}}(\Delta(r_i'))\cap (U_{{N_j'}}(\Delta(r_j'))-(y_F-\Delta(k_i))) \cap\partial W\neq\emptyset \end{align} $$
and
$$ \begin{align} (U_{{N_i'}}(\Delta(r_i'))\cap\partial W)+(y_F-\Delta(k_i))&\subseteq\partial W,\nonumber\\ (U_{{N_j'}}(\Delta(r_j'))\cap\partial W)+(y_F-\Delta(k_j))&\subseteq\partial W. \end{align} $$
Because of equations (17) and (18), the set
$U_{{N'}}(\Delta (r'))\cap U_{{N_i'}}(\Delta (r_i'))\cap (U_{{N_j'}}(\Delta (r_j'))-(y_F-\Delta (k_i)))$
contains a cylinder set
$U_{N}(\Delta (r))$
for which
$U_{N}(\Delta (r))\cap \partial W\neq \emptyset $
and
$$ \begin{align*} \begin{aligned} (U_{{N}}(\Delta(r))\cap\partial W)+(y_F-\Delta(k_i))\subseteq&\ \partial W\quad\text{and}\\ (U_{{N}}(\Delta(r))\cap\partial W)+(y_F-\Delta(k_i))+(y_F-\Delta(k_j))\subseteq&\ \partial W. \end{aligned} \end{align*} $$
Clearly, N can be chosen arbitrarily large.
2.4 Consequences of the weak double disjointness condition (DD′)
The crucial step is now to show that
$\operatorname {int}_{\partial W}(V_k)+(y_F-\Delta (k))\kern1.2pt{\subseteq}\kern1.2pt V_k$
for all
$k\kern1.2pt{\in}\kern1.2pt K$
under suitable assumptions.
Proposition 2.16. Assume that the weak double disjointness condition (DD′) holds. Let the automorphism F of
$(X_\eta ,\sigma )$
be described by a block code
$\{0,1\}^{[-m:m]}\to \{0,1\}$
. Then,
$V_k'+(y_F-\Delta (k))\subseteq V_k'$
for all
$k\in K$
, the set K is contained in
$[-m,m]$
and the sets
$V_k'$
have pairwise disjoint interiors.
Proof. It suffices to prove that
$\operatorname {int}_{\partial W}(V_k)+(y_F-\Delta (k))\subseteq V_k'$
for all
$k\in K$
. Suppose for a contradiction that there is
$k_i\in K$
such that
$\operatorname {int}_{\partial W}(V_{k_i})+(y_F-\Delta (k_i))\not \subseteq V_{k_i}'$
. By Proposition 2.13, there exists
$k_j\in K\setminus \{k_i\}$
such that
$(\operatorname {int}_{\partial W}(V_{k_i})+(y_F-\Delta (k_i)))\cap \operatorname {int}_{\partial W}(V_{k_j})\neq \emptyset $
. So we can apply Corollary 2.15. Hence, there are
$N\in {\mathbb N}$
and
$r\in {\mathbb Z}$
such that, setting
$\beta =y_F-\Delta (k_i)$
,
$$ \begin{align*} (U_N(\Delta(r))\cap \partial W)\subseteq(\partial W-\beta)\cap(\partial W-2\beta-\Delta(k_i-k_j)). \end{align*} $$
In view of condition (DD′), this implies
$k_i=k_j$
. The remaining assertions follow from Corollary 2.14.
Remark 2.17. Suppose that the conclusions of Proposition 2.16 are satisfied (not necessarily condition (DD′)).
-
(a) For
$k\in K$
, let
${\widetilde {\mathcal H}}_n^k:=\{j\in {\widetilde {\mathcal H}}_n: U_n(\Delta (j))\cap V_k'\neq \emptyset \}$
. Then,
${\widetilde {\mathcal H}}_n=\bigcup _{k\in K}{\widetilde {\mathcal H}}_n^k$
by Lemma 2.5 and Corollary 2.14. (However, observe that this need not be a disjoint union, in general!) -
(b)
$j\in {\widetilde {\mathcal H}}_n^k$
if and only if
$j+p_n\in {\widetilde {\mathcal H}}_n^k$
, that is, all
${\widetilde {\mathcal H}}_n^k$
are
$p_n$
-periodic. -
(c) For each
$n>0$
and
$k\in K$
, we have
${\widetilde {\mathcal H}}_n^k+\gcd ((y_F)_n-k,p_n){\mathbb Z}\subseteq {\widetilde {\mathcal H}}_n^k$
.Proof of (c)
For each
$j\in {\widetilde {\mathcal H}}_n^k$
, there exists
$h\in U_n(\Delta (j))\cap V_k'$
. By assumption,
${h+(y_F-\Delta (k)){\mathbb Z}\subseteq V_k'}$
, so that for all
$t\in {\mathbb Z}$
,
$$ \begin{align*} U_n(\Delta(j+((y_F)_n-k)t)\cap V_k'= U_n(\Delta(j)+(y_F-\Delta(k))t)\cap V_k'\neq\emptyset, \end{align*} $$
that is,
$j+((y_F)_n-k){\mathbb Z}\subseteq {\widetilde {\mathcal H}}_n^k$
.
Remark 2.18. Remark 2.17(c) can be used to show that
$|K|=1$
and hence
$\partial W+ (y_F-\Delta (k))\subseteq \partial W$
, whenever
$\boldsymbol {\delta }({\widetilde {\mathcal H}}_n)=o({1}/{\sqrt p_n})$
. This allows us to apply Theorem 2.8, which imposes restrictions on
$y_F$
in terms of the minimal periods
$\tilde {\tau }_n$
of the sets
${\widetilde {\mathcal H}}_n$
. (However, that is quite far from what holds in the
${\mathcal B}$
-free setting.)
Indeed, let
$k,k'\in K$
and denote
$\beta =y_F-\Delta (k)$
and
$\beta '=y_F-\Delta (k')$
. Suppose for a contradiction that
$k\neq k'$
. Then,
$$ \begin{align*} |{\widetilde{\mathcal H}}_n^k\cap[0,p_n)|\ge |\langle\gcd(\beta_n,p_n)\rangle_{p_n}|\quad\text{and}\quad |{\widetilde{\mathcal H}}_n^{k'}\cap[0,p_n)|\ge |\langle\gcd(\beta_n',p_n)\rangle_{p_n}|, \end{align*} $$
where
$\langle s\rangle {p_n}$
denotes the subgroup generated by s in
${\mathbb Z}/p_n{\mathbb Z}$
, so that
$$ \begin{align*} |{\widetilde{\mathcal H}}_n\cap[0,p_n)|^2&\ge |{\widetilde{\mathcal H}}_n^{k}\cap[0,p_n)|\cdot|{\widetilde{\mathcal H}}_n^{k'}\cap[0,p_n)|\\ &\ge |\langle (\gcd(\beta_n,p_n),\gcd(\beta_n',p_n))\rangle_{p_n\times p_n}|, \end{align*} $$
where
$\langle (s,t)\rangle {p_n\times p_n}$
denotes the subgroup generated by
$(s,t)$
in
$({\mathbb Z}/p_n{\mathbb Z})^2$
. Hence,
$$ \begin{align*} \begin{aligned} |{\widetilde{\mathcal H}}_n\cap[0,p_n)|^2 &\ge \operatorname{lcm}\bigg(\frac{p_n}{\gcd(\beta_n,p_n)},\frac{p_n}{\gcd(\beta_n',p_n)}\bigg) =\frac{p_n}{\gcd(\beta_n,\beta_n',p_n)}\\ &\ge \frac{p_n}{\gcd(\beta_n-\beta_n',p_n)} = \frac{p_n}{\gcd(k-k',p_n)}. \end{aligned} \end{align*} $$
However, the last denominator is at most
$2m$
, so
$$ \begin{align*} \frac{1}{2m\cdot p_n}\le \bigg(\frac{|{\widetilde{\mathcal H}}_n\cap[0,p_n)|}{p_n}\bigg)^2 =\boldsymbol{\delta}({\widetilde{\mathcal H}}_n)^2, \end{align*} $$
which is in contradiction to the assumption.
Here is an example of a Toeplitz sequence for which Remark 2.18 applies and for which
$\tau _n$
and
$\tilde {\tau }_n$
, the smallest periods of
${\mathcal H}_n$
and
${\widetilde {\mathcal H}}_n$
, are different.
Example 2.19. Garcia and Hedlund [Reference Garcia and Hedlund14] gave the first example of a
$0$
–
$1$
non-periodic Toeplitz sequence. At each level n of their construction, there is exactly one hole in each interval of length
$p_n$
, so all holes are essential, and the centralizer is trivial because condition (*) is satisfied, see Remark 2.10. Our example is a modification of this construction: (to be precise, the example from [Reference Garcia and Hedlund14] is not really a Toeplitz sequence, because it is not periodic at position
$0$
. However, its orbit closure is minimal and contains many Toeplitz sequences. Our modification takes this into account.)
let
$r_n:=\sum _{j=0}^{n-1}2^{2j}=({2^{2n}-1})/{3}$
. Define a Toeplitz sequence in such a way that
$$ \begin{align*} {\mathcal H}_n=2^{2n}{\mathbb Z}-r_n. \end{align*} $$
Observe that
${\mathcal H}_{n}=2^{2(n-1)}(4{\mathbb Z}-1)-r_{n-1}\subseteq {\mathcal H}_{n-1}$
, in particular,
$\bigcap _{n\ge 1}{\mathcal H}_n=\emptyset $
. Here,
${\mathcal H}_{n-1}\setminus {\mathcal H}_{n}$
is the disjoint union of the residue classes
$2^{2(n-1)}(4{\mathbb Z}-k)-r_{n-1}$
,
$k\in \{0,2,3\}$
, and the positions in each of these residue classes should be filled alternatingly with
$0$
and
$1$
. Then all these positions have minimal period
$2^{2n+1}$
, and
$(p_n)_{n\ge 1}=(2^{2n+1})_{n\ge 1}$
is a period structure for the resulting Toeplitz sequence.
If
$2^{2N}t-r_N\in {\mathcal H}_N$
and
$n>N$
, then
$$ \begin{align*} \begin{aligned} (2^{2N}t-r_N+p_N{\mathbb Z})\cap{\mathcal H}_n &= ((2^{2N}t+(r_n-r_N)+p_N{\mathbb Z})\cap 2^{2n}{\mathbb Z})-r_n\\ &= 2^{2N}\bigg(\bigg(t+\frac{2^{2(n-N)}-1}{3}+2{\mathbb Z}\bigg)\cap 2^{2(n-N)}{\mathbb Z}\bigg)-r_n \end{aligned} \end{align*} $$
is non-empty if and only if t is odd. Hence,
${\widetilde {\mathcal H}}_N=2^{2N}(2{\mathbb Z}+1)-r_N$
and
$\tilde {\tau }_N=2^{2N+1}=p_N=2\tau _N$
. Notice that each interval of length
$p_n$
contains exactly two holes and the distance between them is
${p_n}/{2}$
. However,
$p_{n+1}=4p_n$
, so condition (*) is not satisfied. Nevertheless, the centralizer is trivial by Theorem 2.8 and Remark 2.18.
2.5 Additional arithmetic structure (motivated by the
${\mathcal B}$
-free case)
Throughout this subsection, F is again an automorphism of
$(X_\eta ,\sigma )$
and
$\pi (F(x))=\pi (x)+y_F$
for
$x\in X_\eta $
. We start with a particularly simple situation based on the following (very strong) trivial intersection property: there are
$A_n\subseteq {\mathbb N}$
,
$n\in {\mathbb N}$
such that
$$ \begin{align} &\bigcap_{n\in{\mathbb N}}\langle A_n\rangle=\{0\}\quad\text{and}\quad{\widetilde{\mathcal H}}_n\subseteq{\mathcal M}_{A_n},\; n\in{\mathbb N}. \end{align} $$
We will check this property for some non-trivial
${\mathcal B}$
-free examples, see Examples 3.20, 3.27 and 3.28, and also §3.4.
Proposition 2.20. Suppose property (TI) is satisfied.
-
(a) Then the disjointness condition (D) holds, and there exists a unique
$k\in {\mathbb Z}$
such that
$\partial W+y_F\subseteq \partial W+\Delta (k)$
. -
(b)
$\tilde {\tau }_n\mid (y_F)_n-k$
, where
$\tilde {\tau }_n$
is the minimal period of
${\widetilde {\mathcal H}}_n$
. In particular, if infinitely many
${\widetilde {\mathcal H}}_n$
have minimal period
$p_n$
then
$y_F=\Delta (k)$
.
Proof. (a) If there is some
$y\in \partial W\cap (\partial W-\Delta (k))$
, then
$U_{N}(\Delta (y_N))\cap \partial W=U_{N}(y)\cap \partial W\neq \emptyset $
and
$U_{N}(\Delta (k+y_N))\cap \partial W=U_{N}(\Delta (k)+y)\cap \partial W\neq \emptyset $
, so that
$k=(k+y_N)-y_N\in {\widetilde {\mathcal H}}_N-{\widetilde {\mathcal H}}_N\subseteq {\mathcal M}_{A_N}-{\mathcal M}_{A_N}\subseteq \langle {\mathcal M}_{A_N}\rangle =\langle A_{N}\rangle $
for all
$N>0$
by Lemma 2.5a). Hence,
$k=0$
in view of property (TI). If there are
$y_1,y_2\in \partial W$
and
$k_1,k_2\in {\mathbb Z}$
such that
$y_i+y_F\in \partial W-\Delta (k_i)$
(
$i=1,2$
), then
$U_N(\Delta (k_i+(y_i+y_F)_N))\cap \partial W\neq \emptyset $
(
$i=1,2$
), so that
$k_2-k_1\in \langle A_N\rangle $
for all
$N>0$
as before. Hence,
$k_2=k_1$
because of property (TI).
(b) This follows from part (a) of the lemma and from Theorem 2.8a).
Together with Theorem 2.8, this proposition yields the following corollary.
Corollary 2.21. Suppose that property (TI) is satisfied. If
${M\kern0.8pt{:=}\kern0.8pt\liminf _{n\to \infty }\kern-1pt p_n/\tilde {\tau }_n\kern0.8pt{<}\kern0.8pt\infty }$
, then
$$ \begin{align*}\operatorname{Aut}_\sigma(X_\eta)=\langle \sigma\rangle\oplus\operatorname{Tor},\end{align*} $$
where
$\operatorname {Tor}$
denotes the torsion group of
$\operatorname {Aut}_\sigma (X_\eta )$
. It is a cyclic group (possibly trivial), whose order divides M. In particular, if infinitely many
${\widetilde {\mathcal H}}_n$
have minimal period
$p_n$
, then the centralizer of
$(X_\eta ,\sigma )$
is trivial.
If property (TI) is not satisfied, as is the case for more complex
${\mathcal B}$
-free examples, we need additional tools to verify the assumption in equation (4) of Theorem 2.8. The weak double disjointness condition (DD′) turns out to be instrumental along this way, and hence its verification under mild arithmetic assumptions in Proposition 2.30 will be an important step.
We continue with some arithmetic preparations. For the sake of brevity, we sometimes write
$u\vee v$
instead of
$\operatorname {lcm}(u,v)$
. The following notation will be used repeatedly for positive integers a and k:
$$ \begin{align} a^{\div k}:=\frac{a}{\gcd(a,k)}=\frac{a\vee k}{k}. \end{align} $$
For
$A\subseteq {\mathbb N}$
and
$k\in {\mathbb N}$
, denote
$$ \begin{align} A^{\div k} := \{a^{\div k}: a\in A\}, \end{align} $$
$$ \begin{align} A^{\perp k}:=\{a\in A:\gcd(a,k)=1\} \end{align} $$
and
$$ \begin{align} A^{\textrm{prim}}:=\{a\in A:a'\mid a\Rightarrow a'=a \text{ for all }a'\in A\}. \end{align} $$
If
$A=A^{\textrm {prim}}$
, then A is called primitive.
Remark 2.22. Let
$r,\ell ,s,m\in {\mathbb Z}$
and assume that
$\gcd (m,\ell )\mid s-r$
. Then,
$$ \begin{align} (r+\ell{\mathbb Z})\cap(s+m{\mathbb Z}) = x+(\ell\vee m){\mathbb Z} = g\cdot(\tilde{r}+\tilde{\ell}{\mathbb Z}), \end{align} $$
where
$x{\kern-1.5pt}\in{\kern-1.5pt} \{0,\ldots ,(\ell {\kern-1pt}\vee{\kern-1pt} m){\kern-1pt}-{\kern-1pt}1\}$
is defined uniquely by the first identity,
$g{\kern-1pt}={\kern-1pt}\gcd (x, \ell \vee m)$
,
$\tilde {r}={x}/{g}$
, and
$\tilde {\ell }={\ell \vee m}/{g}$
. Observe that
$\gcd (\tilde {r},\tilde {\ell })=1$
. This formula will be applied in several settings, so that one should keep in mind that
$g,\tilde {r}$
and
$\tilde {\ell }$
depend on
$r,\ell ,s$
and m. For later use, observe also that
$$ \begin{align} g=\gcd(r,\ell)\vee\gcd(s,m). \end{align} $$
Here is the proof of equation (24): as
$x-r\in \ell {\mathbb Z}$
and
$x-s\in m{\mathbb Z}$
, we have
$\gcd (x,\ell )=\gcd (r,\ell )$
and
$\gcd (x,m)=\gcd (s,m)$
. Hence,
$$ \begin{align*} g=\gcd(x,\ell\vee m)=\gcd(x,\ell)\vee\gcd(x,m)=\gcd(r,\ell)\vee\gcd(s,m). \end{align*} $$
We list some further consequences:
$$ \begin{align} \operatorname{lcm}(A^{\div g})=\operatorname{lcm}(A)^{\div g},\quad g\cdot{\mathcal M}_{A^{\div g}}={\mathcal M}_A\cap g{\mathbb Z},\quad\text{and}\quad {\mathcal M}_{A^{\div g}}\subseteq{\mathcal M}_{A^{\div\ell\vee m}}, \end{align} $$
where we used
$\gcd (a,g)\mid \gcd (a,\ell \vee m)$
for the last inclusion. Each subset
$Z\subseteq {\mathbb Z}$
holds
$$ \begin{align} (\tilde{r}+\tilde{\ell}Z)\cap {\mathcal M}_{A} = (\tilde{r}+\tilde{\ell}Z)\cap {\mathcal M}_{A^{\perp{\tilde{\ell}}}}, \end{align} $$
because
$\gcd (\tilde {r},\tilde {\ell })=1$
. (Indeed, if
$a\in A$
,
$z\in Z$
and
$x=\tilde {r}+z\tilde {\ell }\in a{\mathbb Z}$
, then
$\gcd (a,\tilde {\ell })\mid \tilde {r}$
, so that
$\gcd (a,\tilde {\ell })\mid \gcd (\tilde {r},\tilde {\ell })=1$
, i.e.
$a\in A^{\perp {\tilde {\ell }}}$
.) Combining equations (23), (25) and (26) yields
$$ \begin{align} (r+\ell{\mathbb Z})\cap(s+a{\mathbb Z})\cap{\mathcal M}_A = g\cdot((\tilde{r}+\tilde{\ell}{\mathbb Z})\cap{\mathcal M}_{(A^{\div g})^{\perp{\tilde{\ell}}}}). \end{align} $$
Given a set
$A\subseteq {\mathbb Z}$
, we denote by
$\boldsymbol {\delta }(A):=\lim _{N\to \infty }({1}/{\log N})\sum _{k=1}^N({1}/{k})1_A(k)$
the logarithmic density of A (provided the limit exists).
Lemma 2.23. Let
$r,\ell ,s,m\in {\mathbb Z}$
and assume that
$\gcd (m,\ell )\mid s-r$
. Recall that
${\tilde {\ell }={\ell \vee m}/{g}}$
. Then,
$$ \begin{align} \boldsymbol{\delta}((r+\ell{\mathbb Z})\cap(s+m{\mathbb Z})\cap{\mathcal M}_A) = \frac{1}{\ell\vee m}\cdot\boldsymbol{\delta}({\mathcal M}_{(A^{\div g})^{\perp{\tilde{\ell}}}}) \leqslant \frac{1}{\ell\vee m}\cdot\boldsymbol{\delta}({\mathcal M}_{A^{\div\ell\vee m}}), \end{align} $$
$$ \begin{align} \boldsymbol{\delta}((r+\ell{\mathbb Z})\cap(s+m{\mathbb Z})\setminus{\mathcal M}_A) = \frac{1}{\ell\vee m}\cdot(1-\boldsymbol{\delta}({\mathcal M}_{(A^{\div g})^{\perp{\tilde{\ell}}}})). \end{align} $$
Proof. As in [Reference Hall15, Lemma 1.17], we have
$$ \begin{align*} \boldsymbol{\delta}((\tilde{r}+\tilde{\ell}{\mathbb Z})\cap{\mathcal M}_{(A^{\div g})^{\perp{\tilde{\ell}}}}) =\frac{1}{\tilde{\ell}}\cdot\boldsymbol{\delta}({\mathcal M}_{(A^{\div g})^{\perp{\tilde{\ell}}}}). \end{align*} $$
As
$g\cdot \tilde {\ell }=\ell \vee m$
, this together with equation (23) proves the identity in equation (28). As
$\boldsymbol {\delta }((r+\ell {\mathbb Z})\cap (s+m{\mathbb Z}))={1}/{\ell \vee m}$
, equation (29) follows at once. For the inequality in equation (28), observe that
${\mathcal M}_{(A^{\div g})^{\perp {\tilde {\ell }}}}\subseteq {\mathcal M}_{A^{\div g}}\subseteq {\mathcal M}_{A^{\div \ell \vee m}}$
by equation (25).
Lemma 2.24. Assume that
$(r+\ell {\mathbb Z})\cap (s+m{\mathbb Z})\cap [N,\infty )\subseteq {\mathcal M}_C$
for some
$r,\ell ,s, m\in {\mathbb Z}$
satisfying
$\gcd (m,\ell )\mid s-r$
, some
$N\in {\mathbb N}$
and a finite set
$C\subset {\mathbb N}$
. Then,
${c\mid \gcd (r,\ell )\vee \gcd (s,m)}$
for some
$c\in C$
.
Proof. Let
$A:=(C^{\div g})^{\perp \tilde \ell }$
. In view of equation (27), our assumption implies
$(\tilde {r}+\tilde {\ell }{\mathbb Z})\cap [N,\infty )\subseteq {\mathcal M}_{A}={\mathcal M}_{A^{\textrm {prim}}}$
. As
$A^{\textrm {prim}}$
is taut, [Reference Dymek, Kasjan, Kułaga-Przymus and Lemańczyk12, Proposition 4.31] shows that there is
$a\in A^{\textrm {prim}}$
such that
$a\mid \gcd (\tilde {r},\tilde {\ell })=1$
, that is,
$1\in A$
. Hence, there is
$c\in C$
such that
$c\mid g=\gcd (r,\ell )\vee \gcd (s,m)$
.
We will need a more detailed arithmetic characterization of the inclusion from Lemma 2.24 when C is a singleton and
$s=0$
.
Lemma 2.25. Let
$r,\ell ,a,c\in {\mathbb Z}$
satisfying
$\gcd (a,\ell )\mid r$
. Then the following conditions are equivalent:
-
(a)
$(r+\ell {\mathbb Z})\cap a{\mathbb Z}\subseteq c{\mathbb Z}$
; -
(b)
$c\mid \gcd (r,\ell )\vee a$
; -
(c)
$c\mid \ell \vee a$
and
$\gcd (c,\ell )\mid r$
.
Proof. By Lemma 2.24, condition (a) implies condition (b). Conversely, condition (a) follows from condition (b), because
$(r+\ell {\mathbb Z})\cap a{\mathbb Z}\subseteq (\gcd (r,\ell )\vee a){\mathbb Z}$
.
Suppose that conditions (a) and (b) hold. By condition (b), we have
$c\mid \gcd (r,\ell )\vee a\mid \ell \vee a$
. Since
$\gcd (a,\ell )\mid r$
, we have
$(r+\ell {\mathbb Z})\cap a{\mathbb Z}\neq \emptyset $
. So by condition (a), we get
$r\in c{\mathbb Z}+\ell {\mathbb Z}=\gcd (c,\ell ){\mathbb Z}$
. Hence, condition (c) holds.
Finally suppose that condition (c) holds. Then,
$$ \begin{align*} \gcd(c,\gcd(r,\ell)\vee a)&=\gcd(c,r,\ell)\vee\gcd(c,a)\\ &=\gcd(c,\ell)\vee\gcd(c,a)=\gcd(c,\ell\vee a)=c, \end{align*} $$
and condition (b) follows at once.
We will need to know the smallest periods of the difference of the sets of multiples.
Lemma 2.26. Assume that A and C are finite subsets of
${\mathbb N}$
and that the set A is primitive.
-
(a) If
$A=\{a\}$
and
$a\not \in {\mathcal M}_C$
, then
$a\cdot \operatorname {lcm}((C^{\div a})^{\mathrm {prim}})$
is the minimal period of
$a{\mathbb Z}\setminus {\mathcal M}_C$
. -
(b) If the set
$C^{\div a}=\{{c}/{\gcd (a,c)}:c\in C\}\subset {\mathbb N}\setminus \{1\}$
is primitive for every
$a\in A$
, then
$\operatorname {lcm}(A \cup C)$
is the minimal period of
${\mathcal M}_{A}\setminus {\mathcal M}_C$
.
Proof. (a) Let
$a\in {\mathbb N}\setminus {\mathcal M}_C$
. Then,
$\emptyset \neq a{\mathbb Z}\setminus {\mathcal M}_C=a\cdot ({\mathbb Z}\setminus {\mathcal M}_{C^{\div a}})$
, and the claim follows from the observation that
$\operatorname {lcm}((C^{\div a})^{\textrm {prim}})$
is the minimal period of
${\mathcal M}_{C^{\div a}}$
, see [Reference Kasjan, Keller and Lemańczyk18, Lemma 5.1b)].
(b) Let T be the minimal period of
${\mathcal M}_A\setminus {\mathcal M}_C$
. Clearly,
$T|\operatorname {lcm}(A\cup C)$
. Now let
$a\in A$
. Since
$1\not \in C^{\div a}$
,
$a\in {\mathcal M}_A\setminus {\mathcal M}_C$
, so
$a+T{\mathbb Z}\subset {\mathcal M}_A$
. By Lemma 2.24 and primitivity of A, we get
$a|T$
, so that
$T+(a{\mathbb Z}\setminus {\mathcal M}_C)\subset a{\mathbb Z}\setminus {\mathcal M}_C$
for every
$a\in A$
and therefore
$a\cdot \operatorname {lcm}((C^{\div a})^{\textrm {prim}})\mid T$
in view of part (a). Hence,
$\operatorname {lcm}(\{a\}\cup C)=a\operatorname {lcm}(C^{\div a})=a\operatorname {lcm}((C^{\div a})^{\textrm {prim}})\mid T$
for every
$a\in A$
, so that
$\operatorname {lcm}(A\cup C)|T$
.
The following proposition is the ‘multi-tool’ of this section.
Proposition 2.27. Let
$A_n$
and
$S_n$
be sets of positive integers. Let
$b\in {\mathbb Z}$
,
$n\geqslant N\geqslant 0$
,
$a\in A_n$
and
$r\in {\mathbb Z}$
be such that
$$ \begin{align} (r+p_N{\mathbb Z})\cap (a{\mathbb Z})\setminus{\mathcal M}_{S_n}\neq\emptyset \end{align} $$
and
$$ \begin{align} ((r+p_N{\mathbb Z})\cap (a{\mathbb Z})\setminus{\mathcal M}_{S_n})+{b}\subseteq{\mathcal M}_{A_n}. \end{align} $$
Assume that there is a subset
$E_{n,a}$
of
$A_n\setminus \{a\}$
for which
$$ \begin{align} \sum_{a'\in E_{n,a}}\frac{1}{\varphi(a^{{\prime}\div a})}<\frac{1}{p_N}, \end{align} $$
where
$\varphi $
is Euler’s totient function. Then there is
$a'\in A_n\setminus E_{n,a}$
such that
$$ \begin{align} {b}\in\gcd(a',\gcd(r,p_N)\vee a){\mathbb Z}\quad \text{and}\quad \emptyset\neq((r+p_N{\mathbb Z})\cap (a{\mathbb Z})\setminus{\mathcal M}_{S_n})\cap(-{b}+a'{\mathbb Z}). \end{align} $$
If
${b}=\beta _n$
for some
$\beta =\Delta (k)$
with
$|k|<\gcd (a',a)$
for all
$a'\in A_n\setminus E_{n,a}$
, then
$\beta =0$
.
Proof. In view of the assumption in equation (31), we have
$$ \begin{align} \begin{aligned} ((r+p_N{\mathbb Z})\cap (a{\mathbb Z})\setminus{\mathcal M}_{S_n})\subseteq (-{b}+{\mathcal M}_{E_{n,a}})\cup(-{b}+ {\mathcal M}_{D_{n,a}}), \end{aligned} \end{align} $$
where
$D_{n,a}=A_n\setminus E_{n,a}$
. We prove below that equation (32) implies
$$ \begin{align} ((r+p_N{\mathbb Z})\cap(a{\mathbb Z})\setminus{\mathcal M}_{S_n})+{b}\not\subseteq{\mathcal M}_{E_{n,a}}. \end{align} $$
Hence and by equation (34), there is
$a'\in D_{n,a}$
such that equation (33) holds. Hence,
${b}=0$
or
$|{b}|\geqslant \gcd (a',\gcd (r,p_N)\vee a)\geqslant \gcd (a',a)$
. It remains to show that equation (32) implies the assumption in equation (35).
Consider any
$a\in A_n$
for which
$ ((r+p_N{\mathbb Z})\cap (a{\mathbb Z})\setminus {\mathcal M}_{S_n})\neq \emptyset $
. Then
$\gcd (a,p_N)\mid r$
, and Lemma 2.23 implies
$$ \begin{align} \begin{aligned} 0<\boldsymbol{\delta}((r+p_N{\mathbb Z})\cap(a{\mathbb Z})\setminus{\mathcal M}_{S_n}) &= \frac1{p_N\vee a}(1-\boldsymbol{\delta}({\mathcal M}_{(S_n^{\div g_n})^{\perp(p_N^{\div g_n})}})) , \end{aligned} \end{align} $$
where