# Interpolation sets for dynamical systems

By Andreas Koutsogiannis, Anh N. Le, Joel Moreira,
Ronnie Pavlov and Florian K. Richter

###### Abstract

Originating in harmonic analysis, interpolation sets were first studied in
dynamics by Glasner and Weiss in the 1980s. A set $S\subset\mathbb{N}$ is an
interpolation set for a class of topological dynamical systems $\mathcal{C}$
if any bounded sequence on $S$ can be extended to a sequence that arises from
a system in $\mathcal{C}$. In this paper, we provide combinatorial
characterizations of interpolation sets for:

  * •

(totally) minimal systems;

  * •

topologically (weak) mixing systems;

  * •

strictly ergodic systems; and

  * •

zero entropy systems.

Additionally, we prove some results on a slightly different notion, called
weak interpolation sets, for several classes of systems. We also answer a
question of Host, Kra, and Maass concerning the connection between sets of
pointwise recurrence for distal systems and $IP$-sets.

###### Contents

  1. 1 Introduction
    1. 1.1 Totally transitive, weak mixing, and strong mixing
    2. 1.2 Minimal and totally minimal systems
    3. 1.3 Uniquely ergodic and strictly ergodic systems
    4. 1.4 Systems of bounded entropy
  2. 2 Background
    1. 2.1 Some families of subsets of integers
    2. 2.2 Some classes of dynamical systems
    3. 2.3 Equivalence of strong interpolation and weak interpolation for some classes of systems
  3. 3 Totally transitive, weak mixing, and strong mixing systems
  4. 4 Minimal and totally minimal systems
  5. 5 Uniquely ergodic and strictly ergodic systems
  6. 6 Systems of bounded entropy
  7. 7 Interpolation sets for distal systems and connections to pointwise recurrence
  8. 8 Open questions

## 1\. Introduction

Several studies have explored the notion of interpolation/interpolating sets
for classes of sequences arising from dynamical systems [11, 12, 24, 25, 36].
The first such definition is due to Glasner [10]: letting $\mathcal{A}$ be a
subset of $\ell^{\infty}(\mathbb{N})$ – the space of all bounded complex-
valued functions on $\mathbb{N}$ – a set $S\subset\mathbb{N}$ is called an
_interpolation set for $\mathcal{A}$_ if every bounded function $f\colon
S\to\mathbb{C}$ can be extended to a function in $\mathcal{A}$. The collection
of all interpolation sets for $\mathcal{A}$ is denoted by
$\mathcal{I}_{\mathcal{A}}$.

A _topological dynamical system_ is a pair $(X,T)$ where $X$ is compact metric
space and $T:X\to X$ is a continuous map. The system $(X,T)$ is _minimal_ if
for every $x\in X$, the (forward) orbit
$o_{T}(x):=\\{T^{n}x:n\in\mathbb{N}_{0}\\}$ is dense in $X$, where
$\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$. Let
$\mathcal{M}\subset\ell^{\infty}(\mathbb{N})$ denote the class of bounded
sequences $x$ for which the closure of the orbit of $x$ under the left shift
map $\sigma$ forms a minimal dynamical system and let $\mathcal{U}$ be the
smallest norm-closed, translation-invariant subalgebra of
$\ell^{\infty}(\mathbb{N})$ containing $\mathcal{M}$. It is an old question
asked by Furstenberg [7] whether $\mathcal{U}=\ell^{\infty}(\mathbb{N})$.
Glasner and Weiss [12] provided a negative answer to this question (for
$\mathbb{Z}$-subshifts rather than $\mathbb{N}$-subshifts) through an
investigation into the interpolation sets for $\mathcal{U}$. If
$\mathcal{U}=\ell^{\infty}(\mathbb{N})$, every subset of $\mathbb{N}$ would be
an interpolation set for $\mathcal{U}$. However, Glasner and Weiss [12] showed
that an interpolation set for $\mathcal{U}$ cannot be piecewise syndetic (see
Section 2 for definition). Recently, Glasner, Tsankov, Weiss, and Zucker [11]
generalized this result from $\mathbb{Z}$ to arbitrary infinite discrete
groups.

The work of Glasner and Weiss mentioned above was motivated partly by a
classical notion in harmonic analysis called Sidon sets. Letting
$\mathcal{F}\subset\ell^{\infty}(\mathbb{N})$ be all Fourier transforms of
measures on the unit circle, a set $S\subset\mathbb{N}$ is _Sidon_ if it is an
interpolation set for $\mathcal{F}$. Sidon [33, 34] himself showed that every
lacunary set is Sidon.111A set $\\{a_{n}:n\in\mathbb{N}\\}\subset\mathbb{N}$
with $a_{1}<a_{2}<\ldots$ is _lacunary_ if
$\inf_{n\in\mathbb{N}}a_{n+1}/a_{n}>1$. A later result of Drury [3] showed
that the union of two Sidon sets is Sidon.

An important special case of Sidon sets arises when one considers
$\mathcal{F}_{0}$ – the set of Fourier transforms of discrete measures on the
unit circle. The interpolation sets for $\mathcal{F}_{0}$ are called _$I_{0}$
-sets_. $I_{0}$-sets were studied extensively starting in the 1960s [16, 17,
23, 30, 31]. Strzelecki [35] showed that lacunary sets are $I_{0}$-sets,
extending Sidon’s results [33, 34]. Ryll-Nardzewski proved that the union of
an $I_{0}$-set and a finite set is an $I_{0}$-set [31]. Using Hartman-Ryll-
Nardzewski characterization [17], some unions of lacunary sets such as
$\\{2^{n}:n\in\mathbb{N}\\}\cup\\{2^{n}+1:n\in\mathbb{N}\\}$ are $I_{0}$-sets.
However, in contrast to Sidon sets, the union of two $I_{0}$-sets is not
necessarily $I_{0}$, for example,
$\\{2^{n}:n\in\mathbb{N}\\}\cup\\{2^{n}+n:n\in\mathbb{N}\\}$ [31]. It is not
true either that every $I_{0}$-set is a finite union of lacunary sets as seen
in the following example by Grow [14] and Méla [27]:
$\\{3^{n^{2}}+3^{j}:n\geqslant 1,(n-1)^{2}\leqslant j\leqslant n^{2}\\}$.

The building blocks of previously known examples of $I_{0}$-sets are lacunary
sets, a feature that makes them extremely sparse. Therefore, it is natural to
ask whether there exists an $I_{0}$-set that has
subexponential222$\\{a_{n}:n\in\mathbb{N}\\}\subset\mathbb{N}$ with
$a_{1}<a_{2}<\ldots$ is _subexponential_ if $\lim_{n\to\infty}(\log
a_{n})/n=0$. or even polynomial growth. Using a dynamical method, the second
author [24] gave a negative answer to this question. This result was extended
to interpolation sets for nilsequences in [2, 25]. Another connection between
interpolation sets and dynamics was discovered by Griesmer [13], who
established a link between $I_{0}$-sets and Katznelson’s question on the
equivalence of Bohr recurrence and topological recurrence.

Interpolation sets also connect to questions in number theory. Sarnak’s Möbius
disjointness conjecture [32], for example, lies at the intersection of
multiplicative number theory and dynamics and has received a lot of attention
in the last decade (see [5, 22] for surveys on recent developments). The
fourth author showed in [28] that every set of Banach density $0$ is a weak
interpolation set (1.2) for the class of all finite-valued sequences whose
shift closure forms a zero-entropy minimal dynamical system. This implies that
Sarnak’s conjecture fails if one replaces the sequence of integers with any
sequence of zero Banach density. In particular, this disproved a polynomial
analogue of Sarnak’s conjecture posed in [4], which has previously been
addressed independently by Kanigowski, Lemańczyk, and Radziwiłł [20] and Lian,
Shi [26].

Given a rich but scattered literature on interpolation sets in dynamics, the
primary goal of this work is a systematic study of interpolation sets for
dynamical systems. Specifically, we characterize the interpolation sets for
sequences arising from following classes of topological dynamical systems:
systems with various mixing-type properties, minimal and totally minimal
systems, uniquely and strictly ergodic systems, and systems with restricted
entropy. We begin our exploration with precise definitions.

######  Definition 1.1 (Strong interpolation).

Let ${\mathcal{C}}$ be a class of topological dynamical systems. A set
$S\subset\mathbb{N}$ is an _interpolation set for ${\mathcal{C}}$_ if for
every bounded function $f:S\to\mathbb{C}$ there exists a system
$(X,T)\in{\mathcal{C}}$, a transitive333A point $x\in X$ is _transitive_ if
its orbit, $o_{T}(x)=\\{T^{n}x:n\in\mathbb{N}_{0}\\}$, is dense in $X$. point
$x\in X$ and $F\in C(X)$ such that $F(T^{n}x)=f(n)$ for every $n\in S$.

In other words, $S$ is an interpolation set for $\mathcal{C}$ if and only if
$S\in\mathcal{I}_{\mathcal{F}}$ for $\mathcal{F}=\\{n\mapsto
F(T^{n}x):(X,T)\in\mathcal{C},\leavevmode\nobreak\ F\in
C(X),\leavevmode\nobreak\ x\in X\leavevmode\nobreak\ \text{transitive}\\}$. To
differentiate with a weaker notion of interpolation sets we are about to
introduce, occasionally we refer to interpolation sets as _strong
interpolation sets_.

In some situations, it will be useful to have a version of interpolation sets
which works for functions from the set $S$ to a finite set, rather than to
$\mathbb{C}$. The size of the finite set in some sense quantifies the “amount
of independence” to be exhibited. A similar idea was already considered by
Weiss [36, Theorem 8.1], where he proved that if $X$ is a $\\{0,1\\}$-subshift
of positive entropy, then there exists $S\subset\mathbb{N}$ of positive
density such that $X|_{S}=\\{0,1\\}|_{S}$. In that spirit, we make the
following definition.

######  Definition 1.2 (Weak interpolation).

Let ${\mathcal{C}}$ be a class of topological dynamical systems and let
$k\in\mathbb{N}$. A set $S\subset\mathbb{N}$ is a _weak interpolation set of
order $k$ for ${\mathcal{C}}$_ if for every bounded $f:S\to\\{0,\dots,k-1\\}$
there exists a system $(X,T)\in{\mathcal{C}}$, a transitive point $x\in X$ and
$F\in C(X)$ such that $F(T^{n}x)=f(n)$ for every $n\in S$. We say that $S$ is
a _weak interpolation set of all orders_ for $\mathcal{C}$ if it is a weak
interpolation set of order $k$ for $\mathcal{C}$ for all $k\in\mathbb{N}$.

There are some obvious relations between these notions of interpolation sets.
It is immediate from the definition that for any $k^{\prime}\leqslant k$,
every weak interpolation set of order $k$ is a weak interpolation set of order
$k^{\prime}$. Similarly, a strong interpolation set is a weak interpolation
set of all orders.

For many classes of systems, the differences between weak interpolation sets
of different orders are genuine. For example, since there exists a minimal and
uniquely ergodic $\\{0,1\\}$-subshift of positive entropy [15], Weiss’s result
[36, Theorem 8.1] implies that there is a positive density set $S$ which is
weak interpolation of order $2$ for the class of uniquely ergodic systems. On
the other hand, in 1.6 below, we prove that every set of positive Banach
density (such as $S$) cannot be a weak interpolation set of all orders for
uniquely ergodic systems.

In general, it is also false that weak interpolation of all orders implies
strong interpolation. Our 1.9 shows that for the class of finite entropy
systems, all sets (including $\mathbb{N}$ itself) are weak interpolation of
all orders, while 1.8 shows that only sets of zero Banach density are strong
interpolation.

That being said, for some special classes of systems, the concepts “weak
interpolation of order $k$” for a specific value of $k$, “weak interpolation
of all orders,” and “strong interpolation” are all equivalent. These classes
include compact abelian group rotations [17, Theorem 1], nilsystems [25,
Theorem 2.2], and minimal systems (1.5 below). In general, these concepts are
equivalent for the class $\mathcal{C}$ if $\mathcal{C}$ is closed under taking
subsystems and countable Cartesian products (see Section 2.3).

The notions of interpolation sets we discussed above also connect to the
notions of “null” and “tame” systems, which are important classes with several
useful equivalent definitions. _Null systems_ are those with zero topological
sequential entropy for every subsequence, and _tame systems_ are those whose
Ellis enveloping semigroup has cardinality at most $2^{\aleph_{0}}$. For
minimal systems, null is strictly stronger than tame, which is strictly
stronger than uniquely ergodic with (measurable) discrete spectrum (see [19]).
The connection to interpolation sets arises from alternative definitions of
these notions in terms of so-called _independence sets_ (see [21]). For
instance, in [6] it is shown that a system is non-tame if and only if there
exist disjoint compact sets $K_{0},K_{1}$ and an infinite set
$S\subset\mathbb{N}$ so that for every $f:S\rightarrow\\{0,1\\}$, there exists
$x\in X$ with $T^{s}x\in K_{f(s)}$ for all $s\in S$. A system is non-null if
and only if there exist disjoint compact sets $K_{0},K_{1}$ and arbitrarily
large finite sets $S$ with the above properties. These conditions are not
equivalent to ours; for instance, for any non-tame system $(X,T)$, there is an
infinite weak interpolation set $S$ of order $2$ for the singleton class
$\mathcal{C}=\\{(X,T)\\}$ (by using Urysohn’s Lemma with the sets
$K_{0},K_{1}$), but it’s not clear whether the converse is true (since in our
definition, every function from $S$ to $\\{0,1\\}$ could correspond to a
different $F\in C(X)$).

### 1.1. Totally transitive, weak mixing, and strong mixing

Three central notions of mixing in topological dynamical systems are (strong)
mixing, weak mixing and total transitivity (see Section 2 for the
definitions). The following implications are well known: mixing $\Rightarrow$
weak mixing $\Rightarrow$ totally transitive. Therefore, by definition,

interpolation for mixing systems $\displaystyle\Downarrow$ interpolation for
weak mixing systems $\displaystyle\Downarrow$
$\displaystyle\text{interpolation for totally transitive systems}.$

Our first result asserts that the converse directions of both implications
above also hold, and provides an easy-to-check combinatorial description of
interpolation sets for these classes.

######  Theorem 1.3.

Let $S\subset\mathbb{N}$. The following are equivalent:

  1. 1.

$S$ is a weak interpolation set of all orders for totally transitive systems.

  2. 2.

$S$ is an interpolation set for totally transitive systems.

  3. 3.

$S$ is an interpolation set for weak mixing systems.

  4. 4.

$S$ is an interpolation set for mixing systems.

  5. 5.

$S$ is not syndetic, i.e. $S$ has arbitrarily large gaps (see Section 2 for
definition).

We do not know whether weak interpolation of order $2$ for totally transitive
systems is also equivalent to the condition that $S$ is not syndetic; see 8.7.

######  Remark 1.4.

It is natural to wonder why we did not consider interpolation sets for the
class of transitive systems. This is because by our definition, every subset
of $\mathbb{N}$ is such an interpolation set. Indeed, let
$z=(z(n))_{n\in\mathbb{N}_{0}}$ be an arbitrary bounded sequence (function)
and let $X$ be the closure of the orbit of $z$ under the left shift $\sigma$.
By definition, the system $(X,\sigma)$ is transitive with $z$ as a transitive
point. In addition, $F(\sigma^{n}z)=z(n)$ for all $n\in\mathbb{N}$ for the
continuous function $F:X\to\mathbb{C}$ defined by
$F((x(n))_{n\in\mathbb{N}_{0}})=x(0)$.

### 1.2. Minimal and totally minimal systems

Recall that $\mathcal{M}$ denotes the class of bounded sequences whose shift
closure forms a minimal dynamical system and $\mathcal{U}$ the closed, shift
invariant subalgebra of $\ell^{\infty}(\mathbb{N})$ generated by
$\mathcal{M}$. In [12, Theorem 1], Glasner and Weiss showed that $S$ is an
interpolation set for $\mathcal{U}$ if and only if $S$ is not piecewise
syndetic.444See Section 2 for definition; in [12], non-piecewise syndetic sets
are called _small sets_. Since $\mathcal{M}\subset\mathcal{U}$, it follows
that an interpolation set for $\mathcal{M}$ must be non-piecewise syndetic.
However, because the inclusion $\mathcal{M}\subset\mathcal{U}$ is strict, the
converse direction is unknown, i.e., is every non-piecewise syndetic set an
interpolation set for $\mathcal{M}$? Our next theorem confirms that this is
indeed the case. In addition, we show that every non-piecewise syndetic set is
an interpolation set for a smaller class, namely, totally minimal systems.

######  Theorem 1.5.

Let $S\subset\mathbb{N}$. The following are equivalent:

  1. 1.

$S$ is a weak interpolation set of order $2$ for minimal systems.

  2. 2.

$S$ is a weak interpolation set of all orders for minimal systems.

  3. 3.

$S$ is an interpolation set for minimal systems.

  4. 4.

$S$ is an interpolation set for totally minimal systems.

  5. 5.

$S$ is not piecewise syndetic.

### 1.3. Uniquely ergodic and strictly ergodic systems

A topological system $(X,T)$ is _uniquely ergodic_ if there is only one Borel
measure $\mu$ on $X$ that is preserved by $T$ (i.e., $\mu(A)=\mu(T^{-1}A)$ for
every Borel set $A\subset X$), and is _strictly ergodic_ if it is uniquely
ergodic and minimal. The next theorem characterizes interpolation sets for
uniquely ergodic systems as well as for strictly ergodic systems.

######  Theorem 1.6.

Let $S\subset\mathbb{N}$. The following are equivalent:

  1. 1.

$S$ is a weak interpolation set of all orders for uniquely ergodic systems.

  2. 2.

$S$ is an interpolation set for uniquely ergodic systems.

  3. 3.

$S$ is an interpolation set for strictly ergodic systems.

  4. 4.

$S$ has zero Banach density, i.e., $d^{*}(S)=0$ (see Section 2 for
definition).

As we have mentioned before, by Hahn-Katznelson’s [15] and Weiss’s results
[36, Theorem 8.1], there is a set $S$ of positive density such that $S$ is
weak interpolation of order $2$ for the class of strictly ergodic systems.
This fact and 1.6 imply two things: First, for the classes of uniquely ergodic
systems and strictly ergodic systems, weak interpolation of order $2$ and weak
interpolation of all orders are not equivalent. Second, it leads to the
question: Can a weak interpolation set of order $2$ for uniquely ergodic
systems be syndetic? At the moment, we do not know the answer to this question
(see 8.8). However, the answer is negative if we impose an additional
condition on the growth of the corresponding complexity function:

######  Proposition 1.7.

If $S\subset\mathbb{N}$ is syndetic and the word complexity of the sequence
$1_{S}$ grows subexponentially (i.e. the orbit closure of $1_{S}$ under the
left shift has zero entropy), then $S$ is not a weak interpolation set of
order $2$ for uniquely ergodic systems.

### 1.4. Systems of bounded entropy

Topological entropy is a measure of the complexity of a dynamical system.
Since the definition of entropy is rather technical, we delay it until Section
2. It turns out that classifying interpolation sets for the class of systems
whose entropy is bounded by a fixed constant is simple: in the definition of
interpolation sets we allow the function to be extended to take infinitely
many values, and so interpolation sets for finite entropy systems must have
zero Banach density.

######  Proposition 1.8.

Let $S\subset\mathbb{N}$. The followings are equivalent:

  1. 1.

$S$ is an interpolation set for systems of finite topological entropy.

  2. 2.

$S$ is an interpolation for systems of topological entropy $\leqslant M$ for
some fixed $M\geqslant 0$.

  3. 3.

$S$ is an interpolation set for systems of zero topological entropy.

  4. 4.

$d^{*}(S)=0$.

On the other hand, every subset of $\mathbb{N}$ is a weak interpolation set of
all orders for finite entropy systems. This is a consequence of the fact that
the full shift on a finite alphabet has a finite entropy.

######  Proposition 1.9.

Every subset $S\subset\mathbb{N}$ is a weak interpolation set of all orders
for systems of finite entropy.

Two results above show that for the class of finite entropy systems, strong
interpolation and weak interpolation of all orders are not equivalent.

Due to the aforementioned simple characterizations, when working with finite
entropy systems, it is more interesting to consider weak interpolation sets of
a fixed order $k$. In this regard, there is an interesting relation between
the entropy bound and the density of the weak interpolation sets. To state the
next theorem, we define the entropy function $H:[0,1]\to[0,\log 2]$ as

$H(\delta)=\begin{cases}-\delta\log\delta-(1-\delta)\log(1-\delta),\text{ if
}\delta\in(0,1)\\\ 0,\text{ if }\delta=0\text{ or }1.\end{cases}$

######  Theorem 1.10.

Let $\delta\in[0,1/2]$. For any $k\in\mathbb{N}$ and $S\subset\mathbb{N},$ we
have the following:

  1. 1.

If $d^{*}(S)\leqslant\delta$, then $S$ is a weak interpolation set of order
$k$ for systems of entropy $\leqslant H(\delta)+\delta\log(k-1)$.

  2. 2.

If $d^{*}(S)>\delta$, then $S$ is not a weak interpolation set of order $k$
for the class of systems of entropy $\leqslant\delta\log k$.

Moreover, (2) is sharp in the following sense:

  1. 3.

There exists $S\subset\mathbb{N}$ with $d^{*}(S)=\delta$ such that $S$ is a
weak interpolation set of order $k$ for the class of systems of entropy
$\leqslant\delta\log k$.

We do not know whether (1) is sharp, i.e. whether the bound
$H(\delta)+\delta\log(k-1)$ could be improved, and leave this as a question.

Outline of paper. Section 2 contains some background and general discussion.
Sections 3-6 concern, respectively, totally transtive/weak mixing/mixing
systems, minimal systems, strictly ergodic systems, and systems with bounded
entropy. In Section 7 we prove some results about distal systems, including an
answer to a question of Host, Kra, and Maass [18] on the connection between
sets of pointwise recurrence for distal systems and $IP$-sets. Finally,
Section 8 contains some open questions that naturally arise from our study.

## 2\. Background

### 2.1. Some families of subsets of integers

######  Definition 2.1.

Let $S$ be a nonempty subset of $\mathbb{N}$.

  * $\bullet$

$S$ is _syndetic_ if the gaps between consecutive elements of $S$ are bounded.
Equivalently, $S$ is syndetic if there exists $k\in\mathbb{N}$ such that
$S\cup(S-1)\cup\ldots\cup(S-k)\supset\mathbb{N}$.

  * $\bullet$

$S$ is _thick_ if it contains arbitrary long intervals of the form
$\\{m,m+1,\ldots,n\\}$.

  * $\bullet$

$S$ is _piecewise syndetic_ if it is the intersection of a syndetic set and a
thick set.

  * $\bullet$

$S$ is _thickly syndetic_ if for every $n\in\mathbb{N}$, there exists a
syndetic set $E$ such that $\bigcup_{m\in E}\\{m,m+1,\ldots,m+n\\}\subset S$.
Equivalently, $S$ is thickly syndetic if and only if its complement
$\mathbb{N}\backslash S$ is not piecewise syndetic.

The classes of thickly syndetic and piecewise syndetic sets are dual in the
following sense: every thickly syndetic set intersects every piecewise
syndetic set and vice versa.

######  Definition 2.2.

The _upper Banach density of $S\subset\mathbb{N}$_ is

$d^{*}(S)=\sup_{(F_{N})_{N\in\mathbb{N}}}\limsup_{N\to\infty}\frac{|S\cap
F_{N}|}{|F_{N}|}$

where the supremum is taken over all sequences of intervals
$(F_{N})_{N\in\mathbb{N}}$ in $\mathbb{N}$ satisfying $|F_{N}|\to\infty$ as
$N\to\infty$. An equivalent definition of the upper Banach density of $S$ is

$d^{*}(S)=\lim_{n\to\infty}\sup_{m\in\mathbb{N}}\frac{|S\cap\\{m,\ldots,m+n-1\\}|}{n}.$

If $d^{*}(S)=0$, then $\limsup_{N\to\infty}\frac{|S\cap F_{N}|}{|F_{N}|}=0$
for any sequence $(F_{N})_{N\in\mathbb{N}}$ of intervals in $\mathbb{N}$ whose
lengths tend to infinity. In this case, we say $S$ has _zero Banach density_.

### 2.2. Some classes of dynamical systems

######  Definition 2.3.

A _topological dynamical system_ (or a _system_ for short) is a pair $(X,T),$
where $X$ is a compact metric space and $T:X\to X$ is a continuous map.

A point $x\in X$ is _transitive_ if its forward orbit
$o_{T}(x):=\\{T^{n}x:\;n\in\mathbb{N}_{0}\\}$ is dense in $X$.

######  Definition 2.4.

A system $(X,T)$ is

  * $\bullet$

_transitive_ if it contains a transitive point,

  * $\bullet$

_totally transitive_ if $(X,T^{k})$ is transitive for every $k\in\mathbb{N}$,

  * $\bullet$

(topological) _weakly mixing_ if for any open $U,V\subset X$, the set
$\\{n\in\mathbb{N}:\;T^{-n}U\cap V\neq\varnothing\\}$ is thick,

  * $\bullet$

(topological) _mixing_ if for any open $U,V\subset X$, the set
$\\{n\in\mathbb{N}:\;T^{-n}U\cap V\neq\varnothing\\}$ is co-finite,

  * $\bullet$

_uniquely ergodic_ if there exists a unique Borel probability measure $\mu$ on
$X$ satisfying $\mu(A)=\mu(T^{-1}A)$ for every Borel set $A\subset X$,

  * $\bullet$

_totally uniquely ergodic_ if $(X,T^{k})$ is uniquely ergodic for every
$k\in\mathbb{N}$.

######  Remark 2.5.

There are several other notions of transitivity in topological dynamics. Among
them, the most commonly used is probably the following: A system $(X,T)$ is
_set transitive_ if for any open $U,V$, there exists $n\in\mathbb{N}$ such
that $U\cap T^{-n}V\neq\varnothing$; and $(X,T)$ is _set totally transitive_
if $(X,T^{k})$ is set transitive for all $k\in\mathbb{N}$.

When $X$ is a compact metric space, set transitivity implies (point)
transitivity – the notion we introduce in 2.4. Likewise, set total
transitivity implies (point) total transitivity. The converse implications of
both statement also hold if $X$ has no isolated point. (See [1] for a
discussion on various notions of transitivity and their relations.)

######  Definition 2.6.

Let $(X,T)$ be a system and let $\rho$ be a metric on $X$. Let
$\varepsilon>0.$ If $c(n,\varepsilon,T)$ denotes the minimum cardinality of
covers of $X$ of sets with $\rho_{n}$-diameter less than $\varepsilon,$ where
$\rho_{n}(x,y)=\max_{0\leqslant k\leqslant n}\rho(T^{k}x,T^{k}y),$ we define
as _topological entropy_ of the system the (finite) number

$h(T)=\lim_{\varepsilon\to
0^{+}}\lim_{n\to\infty}\frac{\log(c(n,\varepsilon,T))}{n}.$

######  Definition 2.7.

Let $\mathcal{A}$ be a compact set and define
$\Omega_{\mathcal{A}}=\mathcal{A}^{\mathbb{N}_{0}}$ endowed with the product
topology. Every point $x\in\Omega_{\mathcal{A}}$ is a sequence
$(x(n))_{n\in\mathbb{N}_{0}}$ with $x(n)\in\mathcal{A}$. Define the _left
shift_ $\sigma:\Omega_{\mathcal{A}}\to\Omega_{\mathcal{A}}$ by
$\sigma((x(n))_{n\in\mathbb{N}_{0}})=(x(n+1))_{n\in\mathbb{N}_{0}}$.

The dynamical system $(\Omega_{\mathcal{A}},\sigma)$ is mixing and is called
_the full shift on $\mathcal{A}$_. A _subshift on $\mathcal{A}$_ is a pair
$(X,\sigma)$ where $X$ is a closed subset of $X$ satisfying $\sigma(X)\subset
X$.

### 2.3. Equivalence of strong interpolation and weak interpolation for some
classes of systems

In this section, we show the equivalence of various notions of interpolation
sets for certain classes of systems. In the special case that $\mathcal{C}$ is
the class of compact abelian group rotations, this result was proved in [17,
Theorem 1]. An analogous result for nilsystems was stated in [25, Theorem 2.2]
without proof. Our proof here is similar to the one in [17, Theorem 1].

######  Proposition 2.8.

If $\mathcal{C}$ is a class of dynamical systems satisfying

  1. (1)

If $(X,T)\in\mathcal{C}$ and $(Y,T)$ is a subsystem of $(X,T)$, then
$(Y,T)\in\mathcal{C}$ and

  2. (2)

If $(X,T)$ and $(Y,R)\in\mathcal{C}$, then $(X\times Y,T\times
R)\in\mathcal{C}$,

then weak interpolation of order $2$ for $\mathcal{C}$ is equivalent to weak
interpolation of all orders for $\mathcal{C}$. If in addition $\mathcal{C}$
satisfies

  1. (2’)

If $(X_{n},T_{n})\in\mathcal{C}$ for $n\in\mathbb{N}$, then
$(\prod_{n}X_{n},\prod_{n}T_{n})\in\mathcal{C}$,

then weak interpolation of order $2$ for $\mathcal{C}$ is equivalent to strong
interpolation for $\mathcal{C}$.

######  Remark 2.9.

We note that (1) and (2) hold for the classes of expansive systems and systems
of finite entropy, and that (1) and (2’) hold for the classes of systems of
zero entropy, compact abelian group rotations, inverse limits of nilsystems,
and distal systems.

On the other hand, (1) fails for several classes of systems, eg., systems with
positive entropy, transitive systems and (weak) mixing system. Condition (2)
fails for minimal systems and uniquely ergodic systems.

###### Proof.

Assume that $\mathcal{C}$ satisfies (1) and (2). Since weak interpolation of
order $k$ implies weak interpolation of all smaller orders, it suffices to
show that weak interpolation of order $2$ implies weak interpolation of order
$2^{m}$ for all $m$. Suppose that $S\subset\mathbb{N}$ is a weak interpolation
set of order $2$ for $\mathcal{C}$ and let $f:S\to\\{0,1,\ldots,2^{m}-1\\}$ be
arbitrary. We can decompose $f=\sum_{k=0}^{m-1}f_{k}$ with
$f_{k}:S\to\\{0,2^{k}\\}$ by binary expansion. Then, for each $k$, since $S$
is weak interpolation of order $2$, there exists
$(X_{k},T_{k})\in\mathcal{C}$, a continuous function $F_{k}:X_{k}\to[0,1]$ and
a point $x_{k}\in X_{k}$ such that

$F_{k}(T_{k}^{n}x_{k})=f_{k}(n)\text{ for all }n\in S.$

Define the orbit closure $Y$ of the point $x:=(x_{k})_{0\leqslant k<m}$ in the
product system $\prod_{k=0}^{m-1}(X_{k},T_{k})$. Denote
$T=\prod_{k=1}^{\infty}T_{k}$. By (1) and (2), since $(Y,T)$ is a subsystem of
a finite product of $(X_{k},T_{k})$, $(Y,T)\in\mathcal{C}$. Finally, define
the continuous function $F\in C\left(\prod_{k=1}^{\infty}X_{k}\right)$ as
$F(y_{0},\ldots,y_{m-1})=\sum_{k=0}^{m-1}2^{k}F_{k}(y_{k})$ and note that
$F(T^{n}x)=f(n)$ for $n\in S$. This shows that $S$ is weak interpolation of
order $2^{m}$ for $\mathcal{C}$, and since $m$ was arbitrary, of all orders.

The proof of the second part is similar. Again, we need only prove that if
$\mathcal{C}$ satisfies (1) and (2’) and $S\subset\mathbb{N}$ is weak
interpolation of order $2$ for $\mathcal{C}$, then it is interpolation for
$\mathcal{C}$. Consider any such $S$ and any bounded function
$f:S\to\mathbb{C}$. By the previous part of the proof, we know that $S$ is
weak interpolation of all orders. Letting $C:=\|f\|_{\infty}$, we can write
$f=C\left(\sum_{k=1}^{\infty}f_{k}+ig_{k}\right)$ with
$f_{k},g_{k}:S\to\\{0,\pm 2^{-k}\\}$Then we proceed exactly as above, defining
$(X_{k},T_{k}),(Y_{k},S_{k})\in\mathcal{C}$, $x_{k}\in X_{k}$, $y_{k}\in
Y_{k}$, $F_{k}\in C(X_{k})$, and $G_{k}\in C(Y_{k})$ so that

$F_{k}(T_{k}^{n}x_{k})=f_{k}(n),G_{k}(S_{k}^{n}y_{k})=g_{k}(n)\text{ for all
}n\in S.$

Now we again take the orbit closure of the sequence
$((x_{k},y_{k}))_{k\in\mathbb{N}}$ in the infinite product system, which is in
$\mathcal{C}$ by (1) and (2’). The function
$F:\prod_{k=1}^{\infty}X_{k}\to\mathbb{C}$ defined by
$F((x_{k},y_{k}))_{k\in\mathbb{N}})=C\sum_{k=1}^{\infty}(F_{k}(x_{k})+iG_{k}(y_{k}))$
is a uniform limit of continuous functions, and therefore continuous. Just as
before, $F(T^{n}x)=f(n)$ for all $n\in S$, verifying that $S$ is an
interpolation set for $\mathcal{C}$. ∎

## 3\. Totally transitive, weak mixing, and strong mixing systems

In this section we prove 1.3. We start by establishing some properties of
totally transitive points.

######  Definition 3.1.

Given a totally transitive system $(X,T)$, a point $x\in X$ is a _totally
transitive point_ if $\\{T^{kn}x:n\in\mathbb{N}_{0}\\}$ is dense in $X$ for
every $k\in\mathbb{N}$.

A transitive system always has a transitive point, but it is not immediately
obvious that every totally transitive system has a totally transitive point.
Nevertheless, the next lemma shows that this is indeed the case.

######  Lemma 3.2.

Let $(X,T)$ be totally transitive. Then one of the following two mutually
exclusive statements is true.

  1. 1.

$T:X\to X$ is surjective and there is a residual set of totally transitive
points.

  2. 2.

There exists exactly one transitive point $x\in X$, which is an isolated point
of $X$. In this case $x$ is a totally transitive point and
$(X\backslash\\{x\\},T)$ is a totally transitive subsystem of $(X,T)$.

###### Proof.

Suppose $T$ is surjective, i.e. $TX=X$. If $x$ is a transitive point of
$(X,T)$, then

$\overline{\\{T^{n}(Tx):n\in\mathbb{N}_{0}\\}}=T\overline{\\{T^{n}x:n\in\mathbb{N}_{0}\\}}=TX=X,$

and so $Tx$ is also a transitive point. In particular, whenever $(X,T)$ is
transitive and $T$ is surjective, there exists a dense set of transitive
points. Surjectivity of $T$ implies surjectivity of $T^{k}$, so the system
$(X,T^{k})$ contains a dense set of transitive points.

If $x\in X$ is transitive for $T^{k}$, then for every non-empty open set
$U\subset X$ there exists some $n\in\mathbb{N}_{0}$ with $x\in T^{-nk}U$, so
the union $\bigcup_{n\in\mathbb{N}_{0}}T^{-nk}U$ contains every transitive
point and hence is an open dense set. Taking now a countable basis
$(U_{\ell})_{\ell\in\mathbb{N}}$ for $X$, it follows that the intersection
${\mathcal{T}}:=\bigcap_{\ell,k\in\mathbb{N}}\bigcup_{n\in\mathbb{N}_{0}}T^{-kn}U_{\ell}$
is a countable intersection of dense open sets and hence is residual. To
finish the proof, notice that any point in ${\mathcal{T}}$ is totally
transitive, since for every $k\in\mathbb{N}$ and every non-empty open set
$U\subset X$ there exists $\ell\in\mathbb{N}$ with $U_{\ell}\subset U$ and
hence there exists some $n\in\mathbb{N}_{0}$ such that $T^{nk}x\in
U_{\ell}\subset U$.

(2) If $(X,T)$ is a transitive system that is not surjective, then we claim
that $U:=X\backslash TX$ has exactly one element. Indeed, $TX$ is compact,
hence $U$ is open. By transitivity, there exists a point $x\in X$ with a dense
orbit, so $T^{n}x\in U$ for some $n\geqslant 0$; the definition of $U$ forces
$n=0$ and hence $x\in U$. If there were some $y\in U$ with $y\neq x$ we could
take $V\subset U$ to be a neighborhood of $y$ not containing $x$. The orbit of
$x$ never enters $V$ which is a contradiction, and this proves the claim that
$U=\\{x\\}$.

Let $X_{0}=X\backslash\\{x\\}=TX$. Let $k\in\mathbb{N}$ and $y=T^{k}x$. Then
$y\in X_{0}$. Since $x$ is a totally transitive point and $x$ is isolated,

$\overline{\\{T^{kn}y:n\geqslant 0\\}}=\overline{\\{T^{kn}x:n\geqslant
1\\}}=X_{0},$

showing that $y$ is a transitive point for $(X_{0},T^{k})$, and in particular
that $(X_{0},T^{k})$ is a transitive system. Since $k$ is arbitrary,
$(X_{0},T)$ is totally transitive.

Since $X_{0}=TX$, for $k\geqslant 2$, we have $\\{T^{kn}x:n\geqslant
1\\}\subset TX_{0}$. Because the set in the left hand side is dense in
$X_{0}$, $TX_{0}$ is dense in $X_{0}$. As both $TX_{0}$ and $X_{0}$ are
closed, we conclude that $TX_{0}=X_{0}$, i.e. $T$ is surjective on $X_{0}$. ∎

######  Lemma 3.3.

If $(X,T)$ is totally transitive then every transitive point of $(X,T)$ is a
totally transitive point.

###### Proof.

If $T$ is not surjective, then by 3.2, $(X,T)$ has exactly one transitive
point. Since $(X,T)$ is totally transitive, this unique transitive point is
also a totally transitive point.

Next, assume that $T$ is surjective and let $x$ be a transitive point of
$(X,T)$. Let $k\geqslant 2$ and take
$Y=\overline{\\{T^{kn}x:n\in\mathbb{N}_{0}\\}}$. Since $(X,T^{k})$ is
transitive, there exists a point $y\in X$ that is $T^{k}$-transitive. Since
$x$ is transitive, it follows that $Y\cup T^{-1}Y\cup\ldots\cup T^{-(k-1)}Y=X$
and hence $T^{j}y\in Y$ for some $j\in\\{0,1,\ldots,k-1\\}$. Since $y$ is
$T^{k}$-transitive and $T$ is surjective, $T^{j}y$ is also $T^{k}$-transitive.
It follows that $Y$ contains a $T^{k}$-transitive point of $X$, and is
invariant under $T^{k}$, so $Y=X$, proving that $x$ is $T^{k}$-transitive. ∎

The next lemma is the main technical step towards proving 1.3.

######  Lemma 3.4.

Let $S\subset\mathbb{N}$ be a syndetic set and let $k\in\mathbb{N}$ be such
that $S\cup(S-1)\cup\ldots\cup(S-(k-1))\supset\mathbb{N}$. Let $h>k$ and
$S_{i}:=S\cap(h^{2}\mathbb{N}+[ih,(i+1)h))$ for $0\leqslant i\leqslant h-1$.
For any totally transitive system $(X,T)$ and transitive point $x\in X$, there
exists $0\leqslant i<j\leqslant h-1$ such that

$\overline{\\{T^{n}x:n\in S_{i}\\}}\cap\overline{\\{T^{n}x:n\in
S_{j}\\}}\neq\varnothing.$

###### Proof.

Since $S_{i}$ is nonempty for all $i$, the conclusion is obviously true if
$(X,T)$ is the trivial one-point system. Now suppose $(X,T)$ is a nontrivial,
totally transitive system and let $x$ be a transitive point of $(X,T)$. By
Lemma 3.3, the point $x$ is a totally transitive point. We then have that
$T^{a\mathbb{N}+b}x$ is dense in $X\backslash\\{x\\}$ for any
$a,b\in\mathbb{N}$ (where, for any set $S\subset\mathbb{N}_{0}$,
$T^{S}x:=\\{T^{n}x:n\in S\\}$); this can be seen by analysing the two cases
arising in 3.2 individually.

For $0\leqslant i\leqslant h-1$, let $X_{i}:=\overline{T^{S_{i}}x}$. Note that
for all $0\leqslant i\leqslant h-1$ and $0\leqslant j\leqslant k-1$,

$h^{2}\mathbb{N}+[ih,(i+1)h)-j\supset h^{2}\mathbb{N}+ih.$

Therefore, for $0\leqslant i\leqslant h-1$ and $0\leqslant j\leqslant k-1$,

$S_{i}-j=(S\cap(h^{2}\mathbb{N}+[ih,(i+1)h)))-j\supset(S-j)\cap(h^{2}\mathbb{N}+ih).$

And so,

$\displaystyle S_{i}\cup(S_{i}-1)\cup\ldots\cup(S_{i}-(k-1))$
$\displaystyle\supset(S\cup(S-1)\cup\ldots\cup(S-(k-1)))\cap(h^{2}\mathbb{N}+ih)$
$\displaystyle=h^{2}\mathbb{N}+ih.$

It follows that

$\displaystyle X_{i}\cup T^{-1}X_{i}\cup\ldots\cup T^{-(k-1)}X_{i}$
$\displaystyle\supset\overline{T^{S_{i}}x}\cup\overline{T^{S_{i}-1}x}\cup\ldots\cup\overline{T^{S_{i}-(k-1)}x}$
(3.1) $\displaystyle\supset\overline{T^{h^{2}\mathbb{N}+ih}x}$
$\displaystyle\supset X\backslash\\{x\\}.$

Let $\mu$ be a $T$-invariant measure on $X$. Since $(X,T)$ is totally
transitive and nontrivial, it is not a finite system, and since $x$ is a
transitive point, it is not a periodic point. Therefore, $\mu(\\{x\\})=0$ and
$\mu(X\backslash\\{x\\})=1$. Then (3.1) implies $\mu(X_{i})\geqslant 1/k$ for
all $0\leqslant i\leqslant h-1$. Since $h>k$, there must be $0\leqslant
i<j\leqslant h-1$ such that $X_{i}\cap X_{j}\neq\varnothing$ which is the
desired conclusion. ∎

We are ready to prove 1.3 whose statement is repeated here for convenience.

######  1.3.

Let $S\subset\mathbb{N}$. The following are equivalent:

  1. 1.

$S$ is a weak interpolation set of all orders for totally transitive systems.

  2. 2.

$S$ is an interpolation set for totally transitive systems.

  3. 3.

$S$ is an interpolation set for weak mixing systems.

  4. 4.

$S$ is an interpolation set for strong mixing systems.

  5. 5.

$S$ is not syndetic.

###### Proof.

We have the chain of implications among various classes of topological
dynamical systems: Strong mixing $\Rightarrow$ Weak mixing $\Rightarrow$
Totally transitive. Therefore, the following directions are obvious: (4)
$\Rightarrow$ (3) $\Rightarrow$ (2) $\Rightarrow$ (1) and so it remains to
prove (1) $\Rightarrow$ (5) and (5) $\Rightarrow$ (4).

(1) $\Rightarrow$ (5): For the sake of contradiction, suppose $S$ is syndetic.
Let $k\in\mathbb{N}$ be such that $\bigcup_{i=0}^{k-1}(S-i)=\mathbb{N}$. Fix
an arbitrary integer $h>k$ and define
$S_{i}:=S\cap(h^{2}\mathbb{N}+[ih,(i+1)h))$ for $0\leqslant i\leqslant h-1$.
Let $f:S\to\mathbb{R}$ be a bounded function that satisfies $f(n)=i$ for $n\in
S_{i}$.

By Lemma 3.4, for any totally transitive system $(X,T)$ and transitive point
$x\in X$, there exists $0\leqslant i<j\leqslant h-1$ such that
$\overline{T^{S_{i}}x}\cap\overline{T^{S_{j}}x}\neq\varnothing$. Hence, there
does not exist $F\in C(X)$ such that $F(T^{S_{i}}x)=\\{i\\}$ and
$F(T^{S_{j}}x)=\\{j\\}$. As a result, there is no $F\in C(X)$ such that the
function $n\mapsto F(T^{n}x)$ extends $f$. In particular, $S$ is not weak
interpolation of order $h$, and so also not weak interpolation of all orders,
for totally transitive systems.

(5) $\Rightarrow$ (4): Let $S\subset\mathbb{N}$ be a set which is not
syndetic, take an arbitrary bounded function $f:S\to\mathbb{C}$ and let
$K\subset\mathbb{C}$ be a compact set containing $f(S)$. We consider the full
shift with (potentially infinite) alphabet $K$, defined as the topological
dynamical system $(K^{\mathbb{N}_{0}},\sigma)$ where
$\sigma:(z_{n})_{n\in\mathbb{N}_{0}}\mapsto(z_{n+1})_{n\in\mathbb{N}_{0}}$ is
the left shift. Note that $K^{\mathbb{N}_{0}}$, equipped with the product
topology is a compact metric space and that $\sigma$ is continuous. The system
$(K^{\mathbb{N}_{0}},\sigma)$ is strong mixing and, in particular, transitive.
Let $y\in K^{\mathbb{N}_{0}}$ be a transitive point. We think of points in
$K^{\mathbb{N}_{0}}$ as functions $\mathbb{N}_{0}\to K$.

Since $S$ is not syndetic, then for every $n\in\mathbb{N}$ there is an
interval $I_{n}:=\\{m_{n}+1,m_{n}+2,\dots,m_{n}+n\\}$ of length $n$ which is
disjoint from $S$. Extend $f:S\to K$ to a function $g:\mathbb{N}_{0}\to K$ by
letting $g|_{I_{n}}=y|_{\\{1,\dots,n\\}}$. Then the orbit closure of $g$ as an
element of $K^{\mathbb{N}_{0}}$ contains $y$, and since $y$ is a transitive
point, so is $g$. Finally, let $F:K^{\mathbb{N}_{0}}\to\mathbb{C}$ be the
projection onto the $0$-th coordinate. If follows that $F(\sigma^{n}g)=g(n)$
for all $n\in\mathbb{N}_{0}$. In particular, $F(\sigma^{n}g)=f(n)$ for all
$n\in S$, showing that $S$ is an interpolation set for strongly mixing
systems. ∎

## 4\. Minimal and totally minimal systems

In this section we prove 1.5 by combining the next two lemmas. The first lemma
comes from [12].

######  Lemma 4.1 (Glasner-Weiss [12, Theorem 1 (Part 1)]).

Piecewise syndetic sets are not weak interpolation of order $2$ for minimal
systems.

######  Remark 4.2.

The result of Glasner and Weiss given in [12] asserts that any piecewise
syndetic set is not an interpolation set for minimal systems. However, their
proof yields the stronger conclusion after careful examination. Indeed, their
proof involves constructing, for any piecewise syndetic set $S$, a function
$\eta$ with domain $S$ for which $\eta(s)$ cannot be represented as
$F(T^{s}x)$ for $s\in S$ and a minimal $(X,T)$. This function $\eta$ always
has codomain $\\{0,1\\}$, and so this immediately also demonstrates that $S$
is not weak interpolation of order $2$ for minimal systems.

######  Lemma 4.3.

Every non-piecewise syndetic set is an interpolation set for totally minimal
systems.

###### Proof.

Let $S\subset\mathbb{N}$ be a non-piecewise syndetic set. Let
$u:S\to\mathbb{C}$ be an arbitrary bounded function and let
$K\subset\mathbb{C}$ be a compact set containing $u(S)$ and $0$. We construct
a compact set $X\subset K^{\mathbb{N}_{0}}$ iteratively via auxiliary
sequences $m_{k}\in\mathbb{N}$, closed (thereby compact) sets $X_{k}\subset
K^{m_{k}}$ and $X^{\prime}_{k}\subset K^{m_{k}+1}$, and $w_{k}\in X_{k}$.
Define $m_{0}=1$, $X_{0}=K$, $X^{\prime}_{0}=K^{2}$, and $w_{0}=0$. Suppose
now that $k\geqslant 0$ and that $m_{k}$, $X_{k}$, and $w_{k}$ are defined.
Since $X_{k},X^{\prime}_{k}$ are compact, there exist $2^{-k}$-dense finite
subsets $T_{k},T^{\prime}_{k}$ of $X_{k}$ and $X^{\prime}_{k}$ respectively
(in the $\ell^{\infty}$ metric). Without loss of generality, we assume that
$k!$ divides both $|T_{k}|$ and $|T^{\prime}_{k}|$. Define $m_{k+1}$ to be a
multiple of $m_{k}(k+1)!$ with the property that every interval of length
$m_{k+1}$ contains a gap in $S$ of length
$4m_{k}^{2}(|T_{k}|+|T^{\prime}_{k}|)$, i.e. for all $I$ of length $m_{k+1}$,
there exists a subinterval $J$ of length
$4m_{k}^{2}(|T_{k}|+|T^{\prime}_{k}|)$ so that $J\cap S=\varnothing$. (We use
here the fact that $S$ is not piecewise syndetic, which is equivalent to the
fact that for all $n$, gaps of length $n$ occur syndetically in $S$, i.e.
$\mathbb{N}\backslash S$ is thickly syndetic.) Note that by construction, $k!$
divides $m_{k}$ for all $k$.

Define $X_{k+1}$ to be the set of all strings $w$ with length $m_{k+1}$ which
are concatenations of strings in $X_{k}$ and $X^{\prime}_{k}$ and which have
the following property: for each $0\leqslant i<k!$, the set of concatenated
elements of $X_{k}$ whose first letter is at a location in $w$ equal to
$i\pmod{k!}$ is $2^{-k}$-dense in $X_{k}$, and the set of concatenated
elements of $X^{\prime}_{k}$ whose first letter is at a location in $w$ equal
to $i\pmod{k!}$ is $2^{-k}$-dense in $X^{\prime}_{k}$. Similarly define
$X^{\prime}_{k+1}$, with the only change being that elements of
$X^{\prime}_{k+1}$ must have length $m_{k+1}+1$.

We note that both $X_{k+1}$ and $X_{k+1}^{\prime}$ are nonempty. Indeed, if
$T_{k}=\\{w^{(k)}_{1},\ldots,w^{(k)}_{|T_{k}|}\\}$,
$T^{\prime}_{k}=\\{w^{\prime(k)}_{1},\ldots,w^{\prime(k)}_{|T^{\prime}_{k}|}\\}$,
and $u,v$ are arbitrary elements of $X_{k},X^{\prime}_{k}$ respectively, then
the reader may check that the string

$w_{k}w_{k}\ldots w_{k}(w^{(k)}_{1}w^{(k)}_{2}\ldots
w^{(k)}_{|T_{k}|}w^{\prime(k)}_{1}w^{\prime(k)}_{2}\ldots
w^{\prime(k)}_{|T^{\prime}_{k}|}v)\ldots(w^{(k)}_{1}w^{(k)}_{2}\ldots
w^{(k)}_{|T_{k}|}w^{\prime(k)}_{1}w^{\prime(k)}_{2}\ldots
w^{\prime(k)}_{|T^{\prime}_{k}|}v)$

is in $X_{k+1}$, where $w^{(k)}_{1}w^{(k)}_{2}\ldots
w^{(k)}_{|T_{k}|}w^{\prime(k)}_{1}w^{\prime(k)}_{2}\ldots
w^{\prime(k)}_{|T^{\prime}_{k}|}v$ (which has length equal to $1\pmod{k!}$) is
repeated $m_{k}$ times and the number of $w_{k}$ at the beginning is chosen to
make the entire word have length $m_{k+1}$. (Note that the terminal portion
has length
$m_{k}\left(m_{k}|T_{k}|+(m_{k}+1)(|T^{\prime}_{k}|+1)\right)<4m_{k}^{2}(|T_{k}|+|T^{\prime}_{k}|)$,
which is less than $m_{k+1}$.) Similarly,

$w^{\prime(k)}_{1}w_{k}w_{k}\ldots w_{k}(w^{(k)}_{1}w^{(k)}_{2}\ldots
w^{(k)}_{|T_{k}|}w^{\prime(k)}_{1}w^{\prime(k)}_{2}\ldots
w^{\prime(k)}_{|T^{\prime}_{k}|}v)\ldots(w^{(k)}_{1}w^{(k)}_{2}\ldots
w^{(k)}_{|T_{k}|}w^{\prime(k)}_{1}w^{\prime(k)}_{2}\ldots
w^{\prime(k)}_{|T^{\prime}_{k}|}v)$

is in $X^{\prime}_{k+1}$, where the first concatenated $w_{k}$ above is
replaced by $w^{\prime(k)}_{1}$.

Since $w_{k}$ is a prefix of $w_{k+1}$ for all $k$, we can define $y$ to be
the limit of the $w_{k}$, and define $X$ to be the closure of the orbit of
$y$.

We claim that for any $j>0$, $y$ is uniformly recurrent under the shift action
by $\sigma^{j}$. Indeed, for any neighborhood $U$ of $y$, there exists $k>j$
so that $V=\\{z\ :\ \rho(z_{1}\ldots z_{m_{k}},y_{1}\ldots
y_{m_{k}})<2^{-k}\\}\subset U$. (Here $\rho$ denotes the metric in $X_{k}$.)
By definition, $w_{k}=y_{1}\ldots y_{m_{k}}\in X_{k}$. Also by definition, $y$
is a concatenation of elements of $X_{k+1}$ and $X^{\prime}_{k+1}$, meaning
that the set of locations $t$ for which $y(t)\ldots y(t+m_{k+1}-1)\in X_{k}$
or $y(t)\ldots y(t+m_{k+1})\in X^{\prime}_{k}$ is syndetic (with gaps bounded
from above by $m_{k+1}+1$). For each such $t$, there exists a location
$s\equiv-t\pmod{k!}$ so that $\rho(y(t+s)y(t+s+1)\ldots
y(t+s+m_{k}-1),w_{k})<2^{-k}$, i.e. $\sigma^{t+s}y\in V\subset U$. In
addition, $t+s$ is a multiple of $k!$, and so a multiple of $j$. This implies
that the set $\\{n\in\mathbb{N}:\sigma^{jn}y\in U\\}$ is syndetic. Since $U$
was arbitrary, $y$ is uniformly recurrent for $\sigma^{j}$. Since $j$ was
arbitrary, $X$ is totally minimal.

It remains to construct $x_{u}\in X$ for which $x_{u}(s)=u(s)$ for all $s\in
S$. The construction of $x_{u}$ proceeds in steps, where it is continually
assigned values in $K$ on more and more of $\mathbb{N}$, and undefined
portions are labeled by $*$. Formally, define
$x^{(0)}\in(K\cup\\{*\\})^{\mathbb{N}_{0}}$ by $x^{(0)}(s)=u(s)$ for $s\in S$
and $*$ for all other locations. Note (for induction purposes) that $x^{(0)}$
is an infinite concatenation of elements of $X_{0}$ and blocks of $*$ of
length $m_{0}=1$, and that $x^{(0)}$ contains $*$ exactly on the intervals
$(im_{0},(i+1)m_{0}]$ which are disjoint from $S$.

Now, suppose that $x^{(k)}$ has been defined as an infinite concatenation of
elements of $X_{k}$, $X^{\prime}_{k}$, and blocks of $*$ of length $m_{k}$
which contains $*$ on an interval $(im_{k},(i+1)m_{k}]$ if and only if it is
disjoint from $S$. We wish to extend $x^{(k)}$ to $x^{(k+1)}$ by changing some
$*$ symbols to values in $K$. Consider any $i$ for which
$S\cap(im_{k+1},\ldots,(i+1)m_{k+1}]\neq\varnothing$. The portion of $x^{(k)}$
occupying that interval is a concatenation of words in $X_{k}$ and blocks of
$*$ of length $m_{k}$. By definition of $m_{k+1}$,
$(im_{k+1},\ldots,(i+1)m_{k+1}]$ contains a subinterval $J$ of length
$4m_{k}^{2}(|T_{k}|+|T^{\prime}_{k}|)$ which is disjoint from $S$. This must
contain a subinterval of the form $(am_{k},am_{k}+1,\ldots,bm_{k}]$, where
$b-a=m_{k}|T_{k}|+(m_{k}+1)(|T^{\prime}_{k}|+1)$, which is filled with $*$.
Replace this block of $*$ by

$(w^{(k)}_{1}w^{(k)}_{2}\ldots
w^{(k)}_{|T_{k}|}w^{\prime(k)}_{1}w^{\prime(k)}_{2}\ldots
w^{\prime(k)}_{|T^{\prime}_{k}|}v)\ldots(w^{(k)}_{1}w^{(k)}_{2}\ldots
w^{(k)}_{|T_{k}|}w^{\prime(k)}_{1}w^{\prime(k)}_{2}\ldots
w^{\prime(k)}_{|T^{\prime}_{k}|}v),$

where the string $w^{(k)}_{1}w^{(k)}_{2}\ldots
w^{(k)}_{|T_{k}|}w^{\prime(k)}_{1}w^{\prime(k)}_{2}\ldots
w^{\prime(k)}_{|T^{\prime}_{k}|}v$ is repeated $m_{k}$ times. Exactly as
argued above for $w_{k+1}$, this string includes all elements of $T_{k}$ and
$T^{\prime}_{k}$ at locations starting at all residue classes modulo $k!$.
Fill all other blocks of $*$ of length $m_{k}$ in
$x^{(k)}((im_{k+1},\ldots,(i+1)m_{k+1}])$ with arbitrary elements of $X_{k}$.

By definition, this creates a word in $X_{k+1}$, which we denote by
$w^{(k+1)}_{i}$. Define $x^{(k+1)}((im_{k+1},$ $(i+1)m_{k+1}])=w^{(k+1)}_{i}$
for any $i$ as above (i.e. those for which
$S\cap(im_{k+1},\ldots,(i+1)m_{k+1}]\neq\varnothing$) and as $*$ elsewhere.
Note that $x^{(k+1)}$ is an infinite concatenation of words in $X_{k+1}$,
$X^{\prime}_{k+1}$, and blocks of $*$ of length $m_{k+1}$, which contains $*$
on any interval $(im_{k+1},(i+1)m_{k+1}]$ if and only if it is disjoint from
$S$, and which agrees with $x^{(k)}$ on all locations where $x^{(k)}$ did not
contain $*$.

Continue in this way to build a sequence $\left(x^{(k)}\right)$. Since each is
obtained from the previous by changing some $*$s to values in $K$ (which are
not changed in future steps), they approach a limit $x_{u}$. Since
$S\neq\varnothing$, $S\cap(0,m_{k}]\neq\varnothing$ for all large enough $k$,
and so $x^{(k)}((0,m_{k}])$ has no $*$, meaning that $x_{u}\in
K^{\mathbb{N}_{0}}$. By definition, $x_{u}$ agrees with $x^{(0)}$ on all
locations where $x^{(0)}$ did not have $*$, and so $x_{u}(s)=u(s)$ for all
$s\in S$. It remains only to show that $x_{u}\in X$. To see this, we note that
for all sufficiently large $k$, $x_{u}(0,m_{k}]\in X_{k}$ by definition. Also
by definition, for all sufficiently large $k$, $y$ begins with $w_{k+1}$,
which contains a subword within distance $2^{-k}$ of $x_{u}(0,m_{k}]$. This
yields a sequence of shifts of $y$ which converges to $x_{u}$, proving that
$x_{u}\in X$ and $x_{u}(s)=u(s)$ for all $s\in S$.

Now, simply define $F\in C(X)$ by $F(x)=x(0)$. Then for all $s\in S$,
$F(\sigma^{s}x_{u})=x_{u}(s)=u(s)$. Since $u$ was arbitrary and $(X,\sigma)$
is totally minimal, we’ve shown that $S$ is an interpolation set for totally
minimal systems. ∎

######  1.5.

Let $S\subset\mathbb{N}$. The following are equivalent:

  1. 1.

$S$ is a weak interpolation set of order $2$ for minimal systems.

  2. 2.

$S$ is a weak interpolation set of all orders for minimal systems.

  3. 3.

$S$ is an interpolation set for minimal systems.

  4. 4.

$S$ is an interpolation set for totally minimal systems.

  5. 5.

$S$ is not piecewise syndetic.

###### Proof.

The implications $(\ref{item:totally-
minimal})\Rightarrow(\ref{item:minimal})\Rightarrow(\ref{item:weak-all-
order})\Rightarrow(\ref{item:weak-order-2})$ are obvious. The implication
$(\ref{item:not-PW})\Rightarrow(\ref{item:totally-minimal})$ is proved in 4.3
and the implication $(\ref{item:weak-order-2})\Rightarrow(\ref{item:not-PW})$
follows from 4.1. ∎

## 5\. Uniquely ergodic and strictly ergodic systems

We prove 1.6 in this section. First, we observe that if $d^{*}(S)>0$, then $S$
is not a weak interpolation of all orders for uniquely ergodic systems.

######  Lemma 5.1.

If $d^{*}(S)>0$, then $S$ is not a weak interpolation set of all orders for
uniquely ergodic systems.

###### Proof.

Suppose $d^{*}(S)>1/k$ for some $k\in\mathbb{N}$. Let
$(F_{N})_{N\in\mathbb{N}}$ be a sequence of intervals in $\mathbb{N}$ such
that

$\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{n\in F_{N}}1_{S}(n)=d^{*}(S)>1/k.$

Without loss of generality, assume the $F_{N}$’s are pairwise disjoint. Define
the function $f:S\to\\{0,1,\ldots,k-1\\}$ by $f(n)=i$ if $n\in F_{N}$ for some
$N$ satisfying $N\equiv i\bmod k$ and arbitrary if
$n\notin\bigcup_{N\in\mathbb{N}}F_{N}$. We claim that there does not exist a
uniquely ergodic system $(X,T)$, a continuous function $F:X\to\mathbb{C}$, and
a transitive point $x\in X$ such that $f(n)=F(T^{n}x)$ for all $n\in S$. As a
result, $S$ is not a weak interpolation set of all orders for the class of
uniquely ergodic systems.

For the sake of contradiction, assume our claim is false and that such
$(X,T)$, $x$, and $F$ exist. For each $0\leqslant i<k$, define
$X_{i}=F^{-1}(\\{i\\})$; then the sets $X_{i}$ are compact, nonempty, and
disjoint subsets of $X$. We may then, for $0\leqslant i<k$, define nonnegative
continuous functions $g_{i}:X\rightarrow[0,1]$ for which $g_{i}=1$ on $X_{i}$
and $g_{i}=0$ on $X_{j}$ for $j\neq i$. Define $U_{i}=\\{x\ :\ g_{i}(x)>0\\}$;
then by definition the $U_{i}$ are disjoint open sets.

For any fixed $i$ and any $m\in\mathbb{N}$, by definition $\sigma^{s}x\in
X_{i}$ for all $s\in S\cap F_{mk+i}$, and so $g_{i}(\sigma^{s}x)=1$ for all
such $s$. The sequence $(F_{mk+i})_{m\in\mathbb{N}}$ is still a sequence of
intervals in $\mathbb{N}$ whose lengths tend to infinity, and so by unique
ergodicity,

$\lim_{m\rightarrow\infty}\frac{1}{|F_{mk+i}|}\sum_{n\in
F_{mk+i}}g_{i}(\sigma^{n}x)=\int g_{i}\ d\mu.$

We note that $g_{i}=0$ on $U_{i}^{c}$, and so $\int g_{i}\
d\mu\leqslant\mu(U_{i})$. Also, $g_{i}\geqslant 0$ and $g_{i}(\sigma^{s}x)=1$
for $s\in S\cap F_{mk+i}$. This yields

$\lim_{m\rightarrow\infty}\frac{|S\cap F_{mk+i}|}{|F_{mk+i}|}\leqslant\int
g_{i}\ d\mu\leqslant\mu(U_{i}).$

The limit on the left is $d^{*}(S)$, so we get $\mu(U_{i})\geqslant d^{*}(S)$
for all $i$. However, there are $k$ such sets, and $d^{*}(S)>1/k$, so we have
a contradiction. ∎

The next proposition shows that if $d^{*}(S)=0$, then $S$ is an interpolation
set for strictly ergodic systems.

######  Lemma 5.2.

Every set of zero Banach density is an interpolation set for strictly ergodic
systems.

###### Proof.

Let $S\subset\mathbb{N}$ be a set satisfying $d^{*}(S)=0$. Let
$u:S\to\mathbb{C}$ be a bounded function and let $K\subset\mathbb{C}$ be a
compact set containing $u(S)$ and $0$. We construct $X\subset
K^{\mathbb{N}_{0}}$ iteratively via auxiliary sequences $m_{k}\in\mathbb{N}$,
compact sets $X_{k}\subset K^{m_{k}}$, and $w_{k}\in X_{k}$. Define $m_{0}=1$,
$X_{0}=K$, and $w_{0}=0$. Now suppose that $m_{k}$, $X_{k}$, and $w_{k}$ are
defined. Since $X_{k}$ is compact, there exists a $2^{-k}$-spanning subset
$T_{k}$ (in the $\ell^{\infty}$ metric). Since $d^{*}(S)=0$, we can define
$m_{k+1}$ to be a multiple of $(2k+2)m_{k}$ larger than $(2k+2)m_{k}|T_{k}|$
with the property that for every interval $I\subset\mathbb{N}$ of length
$m_{k+1}$, $\frac{|I\cap S|}{|I|}<((2k+2)m_{k})^{-1}$.

Define $X_{k+1}$ to be the set of all concatenations of sets of
$\frac{m_{k+1}}{m_{k}}$ strings in $X_{k}$ with the following two properties:
each $w\in T_{k}$ appears at least once, and $w_{k}$ appears at least
$(1-(k+1)^{-1})\frac{m_{k+1}}{m_{k}}$ times. (This set is nonempty since
$\frac{m_{k+1}}{m_{k}}\geqslant(k+1)|T_{k}|$.) Define $w_{k+1}$ to be any
element in $X_{k+1}$ with $w_{k}$ as a prefix. Since $w_{k}$ is a prefix of
$w_{k+1}$ for all $k$, we can define $y$ to be the limit of the $w_{k}$, and
define $X$ to be the closure of the orbit of $y$.

We claim that $y$ is uniformly recurrent. Indeed, for any neighborhood $U$ of
$y$, there exists $k$ so that $\\{z\ :\ \rho(z_{1}\ldots z_{m_{k}},y_{1}\ldots
y_{m_{k}})<2^{-k}\\}\subset U$. (Here $\rho$ is the metric on $X_{k}$.) By
definition, $w_{k}=y_{1}\ldots y_{m_{k}}\in X_{k}$, and so there exists $v\in
T_{k}$ within distance $2^{-k}$ of $w_{k}$. Finally, $y$ is a concatenation of
elements in $X_{k+1}$, each of which contains $v$ by definition. So, the set
of shifts of $y$ in $U$ is syndetic with gaps bounded by $2m_{k+1}$. Since $U$
was arbitrary, $y$ is uniformly recurrent, and so $X$ is minimal.

Next, we claim that $X$ is uniquely ergodic. First, we note that every $x\in
X$ is a shift of an infinite concatenation of $A_{k}$-words for each $k$, and
that each $A_{k}$-word is a concatenation of $A_{k-1}$-words in which at most
a proportion of $1/k$ are not $w_{k-1}$. Therefore, if $z\in\mathbb{C}$ does
not occur in $w_{k-1}$, the frequency of occurrences of $z$ in $x$ is at most
$1/k$. It follows that if $z$ does not occur in any $w_{k}$, then $z$ has
density $0$ of occurrences in $x$. Since $x$ was arbitrary, this means by the
ergodic theorem that $\mu([z])=0$ for all invariant measures $\mu$. (Here
$[z]$ is the cylinder set $\\{x\in X:x(0)=z\\}$.)

Define $W$ to be the countable set of all complex numbers appearing in some
$w_{k}$, and consider any cylinder of the form
$R=\prod_{i=1}^{m}([a_{i},b_{i}]\times[c_{i},d_{i}])\times\prod_{m+1}^{\infty}K$,
where all $a_{i},b_{i},c_{i},d_{i}\notin W$. Choose any ergodic measure $\mu$
and any $x\in X$ generic555In a system $(X,T)$, a point $x\in X$ is said to be
_generic_ for an ergodic measure $\mu$ if
$\frac{1}{N}\sum_{n=1}^{N}F(T^{n}x)\to\int_{X}F\ d\mu$ for every continuous
function $F:X\to\mathbb{C}$. for $\mu$. Note that since all endpoints are not
in $W$, by earlier observation the boundary of $R$ has $\mu$-measure $0$, and
so

$\mu(R)=\lim_{n\rightarrow\infty}\frac{1}{n}|\\{0\leqslant i<n\ :\
\sigma^{i}x\in R\\}|.$

Consider now $k\in\mathbb{N}$ such that $m_{k}>m$. We note that $y$ is an
infinite concatenation of elements in $X_{k+1}$, and since $X_{k+1}$ is
closed, the same is true for $x$; therefore $x((im_{k+1},(i+1)m_{k+1}])\in
X_{k+1}$ for all $k$. We denote

$N(w_{k})=|\\{0<j\leqslant m_{k}-m\ :\ w_{k}(j)\ldots
w_{k}(j+m-1)\in\prod_{i=1}^{m}([a_{i},b_{i}]\times[c_{i},d_{i}])\\}|.$

Then, since each word $x([im_{k+1},(i+1)m_{k+1}))$ is a concatenation of
elements of $X_{k}$ containing at least $(1-(k+1)^{-1})\frac{m_{k+1}}{m_{k}}$
occurrences of $w_{k}$, we see that

$iN(w_{k})(1-(k+1)^{-1})\frac{m_{k+1}}{m_{k}}\leqslant|\\{0\leqslant
j\leqslant im_{k+1}-m\ :\ \sigma^{j}x\in R\\}|\\\ \leqslant
iN(w_{k})(1-(k+1)^{-1})\frac{m_{k+1}}{m_{k}}+i(k+1)^{-1}m_{k+1}+im_{k+1}\frac{m}{m_{k}}.$

Here, the first term of the upper bound comes from visits to $R$ within one of
the $w_{k}$ concatenated within some $x((jm_{k+1},(j+1)m_{k+1}])\in X_{k+1}$,
and the remaining terms are just an upper bound on the number of possible
visits to $R$ at other locations. By dividing by $L=im_{k+1}$ and taking
limits as $i\rightarrow\infty$ (and recalling that $x$ is generic for $\mu$),
we get

$(1-(k+1)^{-1})\frac{N(w_{k})}{m_{k}}\leqslant\mu(R)\leqslant(1-(k+1)^{-1})\frac{N(w_{k})}{m_{k}}+(k+1)^{-1}+\frac{m}{m_{k}}.$

This estimate holds for all $k$, and so we can let $k\rightarrow\infty$ to see
that $\mu(R)=\lim_{k}\frac{N(w_{k})}{m_{k}}$ (and particular this limit
exists). Since this quantity depends only on the sequence $w_{k}$, $\mu(R)$
does not depend on $x$. Since the collection of cylinders $R$ generates the
Borel $\sigma$-algebra, $\mu$ does not depend on $x$, and so $X$ is uniquely
ergodic.

It remains to construct $x_{u}\in X$ for which $x(s)=u(s)$ for all $s\in S$.
The construction of $x_{u}$ proceeds in steps, where it is continually
assigned values in $K$ on more and more of $\mathbb{N}$, and undefined
portions are labeled by $*$. Formally, define
$x^{(0)}\in(K\cup\\{*\\})^{\mathbb{N}_{0}}$ by $x^{(0)}(s)=u(s)$ for $s\in S$
and $*$ for all other locations. Note (for induction purposes) that $x^{(0)}$
is an infinite concatenation of elements of $X_{0}$ and blocks of $*$ of
length $m_{0}=1$, and that $x^{(0)}$ consists of $*$ on any “interval”
$(im_{0},(i+1)m_{0}]$ which is disjoint from $S$.

Now, suppose that $x^{(k)}$ has been defined as an infinite concatenation of
elements of $X_{k}$ and blocks of $*$ of length $m_{k}$ which consists of only
$*$ on any interval $(im_{k},(i+1)m_{k}]$ which is disjoint from $S$. We wish
to extend $x^{(k)}$ to $x^{(k+1)}$ by changing some $*$ symbols to values in
$K$. Consider any $i$ for which
$S\cap(im_{k+1},\ldots,(i+1)m_{k+1}]\neq\varnothing$. The portion of $x^{(k)}$
occupying that interval is a concatenation of elements of $X_{k}$ and blocks
of $*$ of length $m_{k}$. By definition of $m_{k+1}$,
$|S\cap(im_{k+1},\ldots,(i+1)m_{k+1}]|<\frac{m_{k+1}}{(2k+2)m_{k}}$. This
means that if $x^{(k)}((im_{k+1},\ldots,(i+1)m_{k+1}])$ is written as a
concatenation of elements of $X_{k}$ and blocks of $*$ of length $m_{k}$, at
least $(1-(2k+2)^{-1})\frac{m_{k+1}}{m_{k}}$ are blocks of $*$. Overwrite any
$(1-(k+1)^{-1})\frac{m_{k+1}}{m_{k}}$ of these with $w_{k}$, leaving at least
$\frac{m_{k+1}}{(2k+2)m_{k}}\geqslant|T_{k}|$. Fill all remaining ones with
elements of $T_{k}$ in a way that each is used at least once.

By definition, this creates a word in $X_{k+1}$, which we denote by
$w^{(k+1)}_{i}$. Define $x^{(k+1)}((im_{k+1},$ $(i+1)m_{k+1}])=w^{(k+1)}_{i}$
for any $i$ as above (i.e. those for which
$S\cap(im_{k+1},\ldots,(i+1)m_{k+1}]\neq\varnothing$) and as $*$ elsewhere.
Note that $x^{(k+1)}$ is an infinite concatenation of elements of $X_{k+1}$
and blocks of $*$ of length $m_{k+1}$ which contains $*$ on any interval
$(im_{k+1},(i+1)m_{k+1}]$ which is disjoint from $S$, and that $x^{(k+1)}$
agrees with $x^{(k)}$ on all locations where $x^{(k)}$ did not contain $*$.

Continue in this way to build a sequence $x^{(k)}$. Since each is obtained
from the previous by changing some $*$s to values in $K$ (which are not
changed in future steps), they approach a limit $x_{u}$. Since
$S\neq\varnothing$, $S\cap(0,m_{k}]\neq\varnothing$ for all large enough $k$,
and so $x^{(k)}((0,m_{k}])$ has no $*$, meaning that $x_{u}\in
K^{\mathbb{N}_{0}}$. By definition, $x_{u}$ agrees with $x^{(0)}$ on all
locations where $x^{(0)}$ did not have $*$, and so $x_{u}(s)=u(s)$ for all
$s\in S$. It remains only to show that $x_{u}\in X$. To see this, we note that
for all sufficiently large $k$, $x_{u}(0,m_{k}]\in X_{k}$ by definition. Also
by definition, for all sufficiently large $k$, $y$ begins with $w_{k+1}$,
which contains a subword in $T_{k}$ which is within distance $2^{-k}$ of
$x_{u}(0,m_{k}]$. This yields a sequence of shifts of $y$ which converges to
$x_{u}$, proving that $x_{u}\in X$ and $x_{u}(s)=u(s)$ for all $s\in S$.
Finally, this implies that $S$ is interpolation for strictly ergodic systems
by considering $F\in C(X)$ defined by $F(x)=x(0)$. ∎

We are ready to prove 1.6 whose statement is repeated here for convenience.

######  1.6.

Let $S\subset\mathbb{N}$. The following are equivalent:

  1. 1.

$S$ is a weak interpolation set of all orders for uniquely ergodic systems.

  2. 2.

$S$ is an interpolation set for uniquely ergodic systems.

  3. 3.

$S$ is an interpolation set for strictly ergodic systems.

  4. 4.

$d^{*}(S)=0$.

###### Proof.

It is obvious that $(\ref{item:strong-strict-
ergodic})\Rightarrow(\ref{item:strong-unique-
ergodic})\Rightarrow(\ref{item:weak-unique-ergodic})$. The implication
$(\ref{item:unique-ergodic-zero-banach-density})\Rightarrow(\ref{item:strong-
strict-ergodic})$ follows from 5.2 and $(\ref{item:weak-unique-
ergodic})\Rightarrow(\ref{item:unique-ergodic-zero-banach-density})$ follows
from 5.1. ∎

Lastly, we prove 1.7; we recall the statement here for convenience.

######  1.7.

If $S\subset\mathbb{N}$ is syndetic and the word complexity of the sequence
$1_{S}$ grows subexponentially (i.e. the orbit closure of $1_{S}$ under the
left shift has zero entropy), then $S$ is not a weak interpolation set of
order $2$ for uniquely ergodic systems.

###### Proof.

Suppose for a contradiction that a set $S$ exists which is a weak
interpolation set of order $2$ for uniquely ergodic systems and which is
syndetic and has subexponential word complexity. Say that $S$ is syndetic with
gaps less than $G$, and then for every interval $I$ of length at least $2G$,
$|S\cap I|>(2G)^{-1}|I|$.

For any $n>2G$, denote by $p(n)$ the word complexity of $\chi_{S}$, i.e. the
number of different sets which occur as $(S\cap[k,k+n))-k$ for
$k\in\mathbb{N}$. By subadditivity of upper density, there exists such a set
$J=\\{j_{1},\ldots,j_{|J|}\\}$ which occurs with upper density at least
$(p(n))^{-1}$, i.e. $A=\\{k\ :\ (S\cap[k,k+n))-k=J\\}$ has upper density at
least $(p(n))^{-1}$. By partitioning $A$ by residue classes modulo $n$, we can
pass to a subset $B$ of $A$ with gaps of at least $n$ and upper density at
least $(np(n))^{-1}$. It is easy to partition $B$ into subsets $B_{i}$,
$1\leqslant i\leqslant 2^{|J|}$, with the same upper density. Note that
$|J|>(4G)^{-1}n$ by definition of $N$.

Enumerate the words in $\\{0,1\\}^{|J|}$ as $w_{1},\ldots,w_{2^{|J|}}$. Define
$y:S\rightarrow\\{0,1\\}$ by assigning, for every $1\leqslant i\leqslant
2^{|J|}$, every $m\in B_{i}$, and every $1\leqslant k\leqslant|J|$,
$y(m+j_{k})=w_{i}(k)$, and defining $y$ arbitrarily otherwise. This is well-
defined since we assumed $B$ to have gaps of at least $n\geqslant|J|$. By
assumption, there exists a uniquely ergodic system $(X,T)$, transitive point
$x$, and $f\in C(X)$ so that $f(T^{x}x)=y(s)$ for all $s\in S$.

Denote the unique measure of $X$ by $\mu$. For any $1\leqslant i\leqslant
2^{|J|}$, any $m\in B_{i}$, and any $1\leqslant k\leqslant|J|$,
$f(\sigma^{m+j_{k}}x)=y(m+j_{k})=w_{i}(k)$. Therefore, for each such $m$,
$\sigma^{m}(x)\in\bigcap_{k=1}^{|J|}\sigma^{-j_{k}}f^{-1}\\{w_{i}(k)\\}$;
denote the latter set by $C_{i}$. By unique ergodicity and the fact that
$B_{i}$ has upper density at least $(np(n))^{-1}$,
$\mu(C_{i})\geqslant(np(n))^{-1}$. However, the sets $C_{i}$ are disjoint, so
$2^{|J|}(np(n))^{-1}\leqslant 1$. Since $|J|>(4G)^{-1}n$, we contradict
subexponential growth of $p(n)$ for large enough $n$. Our original hypothesis
was false, and $S$ is not a weak interpolation set of order $2$ for uniquely
ergodic systems. ∎

## 6\. Systems of bounded entropy

######  1.8.

Let $S\subset\mathbb{N}$. The following are equivalent:

  1. 1.

$S$ is an interpolation set for systems of finite entropy.

  2. 2.

$S$ is an interpolation set for systems of entropy $\leqslant M$ for some
fixed $M\geqslant 0$.

  3. 3.

$S$ is an interpolation set for systems of zero entropy.

  4. 4.

$d^{*}(S)=0$.

###### Proof.

It is obvious that (3) $\Rightarrow$ (2) $\Rightarrow$ (1) and so it remains
to prove (4) $\Rightarrow$ (3) and (1) $\Rightarrow$ (4).

(4) $\Rightarrow$ (3): Consider any set $S$ with $d^{*}(S)=0$ and any bounded
$f:S\rightarrow\mathbb{C}$. Extend $f$ to
$\tilde{f}:\mathbb{N}_{0}\rightarrow\mathbb{C}$ by defining $\tilde{f}=0$ on
$\mathbb{N}_{0}\backslash S$. Now, the orbit closure of $\tilde{f}$ under the
left shift $\sigma$ is a shift system $(X,\sigma)$ with
$X\subset\mathbb{C}^{\mathbb{N}_{0}}$. We claim that $(X,\sigma)$ has zero
entropy. Defining $F(x)=x(0)$ for all $x$ we have that for $n\in S$,
$F(\sigma^{n}\tilde{f})=\tilde{f}(n)=f(n)$, and $\tilde{f}$ is a transitive
point in $X$ by definition, so assuming the claim we have that $S$ is an
interpolation set for systems of zero entropy.

We claim that every $x\in X$ satisfies
$d^{*}(x^{-1}(\mathbb{C}\backslash\\{0\\}))=0$. Indeed, for any
$\varepsilon>0$, by definition of upper Banach density, there exists $N$ so
that for every $i\in\mathbb{N}$,

$\frac{|\\{0\leqslant j<N\ :\ \tilde{f}(i+j)\neq 0\\}|}{N}<\varepsilon.$

This property is preserved under shifts and limits, and so holds for any $x\in
X$ as well, implying that $d^{*}(x^{-1}(\mathbb{C}\backslash\\{0\\}))=0$.
Therefore, by the ergodic theorem, the only invariant measure on $(X,\sigma)$
is the delta-measure for the constant sequence $0$. By the variational
principle, $(X,\sigma)$ has topological entropy $0$.

(1) $\Rightarrow$ (4): Consider any set $S$ with $d^{*}(S)>0$ and assume for a
contradiction that $S$ is an interpolation set for the class of finite entropy
systems. Choose $\delta<d^{*}(S)$.

There exist arbitrarily large $m\in\mathbb{N}$ and sets $S_{m}\subset[1,m]$
with $|S_{m}|\geqslant\delta m$ for which infinitely many shifts of $S_{m}$
are contained in $S$. Write $S_{m}=\\{s_{m,1},\ldots,s_{m,|S_{m}|}\\}$ in
increasing order for all $m$. Let $\mathcal{S}^{1}$ denote the unit circle
$\\{z\in\mathbb{C}:|z|=1\\}$. Construct a sequence $f:S\to\mathcal{S}^{1}$
which, for all $k$ and $m$, contains every possible sequence of $k$th roots of
unity of length $|S_{m}|$ along some shift of $S_{m}$, i.e. for any such
string $c=(c_{1},\ldots,c_{|S_{m}|})$, there exists $j_{c}$ so that
$f(j_{c}+s_{i})=c_{i}$ for $1\leqslant i\leqslant|S_{m}|$. By assumption,
there exists a finite entropy system $(X,T)$, $x\in X$, and $F\in C(X)$ where
$F(T^{n}x)=f(n)$ for all $n$. Say that $h(X,T)<\delta\log k$. Now, define
$\varepsilon$ so that points of $X$ within distance $\varepsilon$ cannot have
$F$-values which are distinct $k$th roots of unity. For every $m$ and every
$|S_{m}|$-tuple $c=(c_{1},\ldots,c_{m})$ of $k$th roots of unity,
$F(\sigma^{j_{c}+s_{i}}x)=f(j_{c}+s_{i})=c_{i}$ for $1\leqslant
i\leqslant|S_{m}|$. This implies that the collection $\\{\sigma^{j_{c}}x\\}$
of cardinality $k^{|S_{m}|}\geqslant k^{\delta m}$ is
$(m,\varepsilon)$-separated, and since $m$ can be arbitrarily large,
$h(X,T)\geqslant\delta\log k$, a contradiction. ∎

######  1.9.

Every subset $S\subset\mathbb{N}$ is a weak interpolation set of all orders
for systems of finite entropy.

###### Proof.

We will prove that $\mathbb{N}$ itself is a weak interpolation set of all
orders for the class of systems of finite topological entropy. Then, clearly,
the same will be true for all subsets of $\mathbb{N}.$

Consider any $k$ and any function
$f:\mathbb{N}_{0}\rightarrow\\{0,\ldots,k-1\\}$. The orbit closure of $f$
under the left shift is a subshift $(X,\sigma)$ that has entropy bounded by
$\log k$. Define $F:X\to\\{0,\ldots,k-1\\}$ by $F(x)=x(0)$ for all $x\in X$.
It follows that $f$ is a transitive point of $(X,\sigma)$ and
$F(\sigma^{n}f)=f(n)$ for all $n\in\mathbb{N}$. ∎

Recall the entropy function $H:[0,1]\to\mathbb{R}$ with
$H(\delta)=-\big{(}\delta\log\delta+(1-\delta)\log(1-\delta)\big{)}$ for
$\delta\in(0,1)$ and $H(0)=H(1)=0$. The function $H$ is known to be the
exponential growth rate of the number of words with proportion of fewer than
$\delta$ non-$0$ symbols on a $2$-letter alphabet; we include the proof of the
following corresponding estimate on larger alphabets for completeness.

######  Lemma 6.1.

Fix $k\geqslant 2$. For each $m\in\mathbb{N}$ and $\delta\in[0,1/2]$, let

$S(m,\delta,k):=\Big{\\{}f:[m]\to\\{0,\dots,k-1\\}:\big{|}f^{-1}(0)\big{|}\geqslant(1-\delta)m\Big{\\}}.$

Then

$\displaystyle\lim_{m\to\infty}\frac{\log\big{|}S(m,\delta,k)\big{|}}{m}=\inf_{m}\frac{\log\big{|}S(m,\delta,k)\big{|}}{m}=H(\delta)+\delta\log(k-1).$

###### Proof.

Every element of $S(m,\delta,k)$ can be described by first describing the
locations of the non-$0$ letters, and then assigning any non-$0$ letter at
each of those locations. Therefore,

$|S(m,\delta,2)|=\sum_{i=0}^{\lfloor\delta m\rfloor}(k-1)^{i}\binom{m}{i}.$

Since $\delta\leqslant 1/2$, $\binom{m}{i}$ is increasing from $i=0$ to
$\lfloor\delta m\rfloor$ (and $(k-1)^{i}$ is increasing for all $i$), and so

$(k-1)^{\lfloor\delta m\rfloor}{\binom{m}{\lfloor\delta
m\rfloor}}\leqslant|S(m,\delta,2)|\leqslant(\lfloor\delta
m\rfloor+1)(k-1)^{\lfloor\delta m\rfloor}{\binom{m}{\lfloor\delta m\rfloor}}.$
(6.1)

Define $\delta_{m}=\frac{\lfloor\delta m\rfloor}{m}$. Clearly
$\delta_{m}\leqslant 1/2$ and $\delta_{m}\rightarrow\delta$. By Stirling’s
approximation,

$\displaystyle(k-1)^{\lfloor\delta m\rfloor}\log{\binom{m}{\lfloor\delta
m\rfloor}}$ $\displaystyle=(k-1)^{\delta_{m}m}\log{\binom{m}{\delta_{m}m}}$
$\displaystyle=\delta_{m}m\log(k-1)+\log
m!-\log(\delta_{m}m)!-\log((1-\delta_{m})m)!$
$\displaystyle=\delta_{m}m\log(k-1)+m\log m-m-(\delta_{m}m)\log(\delta_{m}m)$
$\displaystyle+\delta_{m}m-((1-\delta_{m})m)\log((1-\delta_{m})m)+(1-\delta_{m})m+o(m)$
$\displaystyle=\delta_{m}m\log(k-1)+mH(\delta_{m})+o(1).$

Since $H$ is a continuous function and $\delta_{m}\rightarrow\delta$, by
(6.1), $\frac{\log|S(m,\delta,k)|}{m}\rightarrow H(\delta)+\delta\log(k-1)$.

Finally, it’s easily checked that the sequence $\log|S(m,\delta,k)|$ is
subadditive in $k$, and so by Fekete’s subadditivity lemma, the limit of
$\frac{\log|S(m,\delta,k)|}{m}$ is also the infimum. ∎

We recall 1.10 from the introduction.

######  1.10.

Let $\delta\in[0,1/2]$. For any $k\in\mathbb{N}$ and a set
$S\subset\mathbb{N},$ we have the following:

  1. 1.

If $d^{*}(S)>\delta$, then $S$ is not a weak interpolation set of order $k$
for systems of entropy $\leqslant\delta\log k$.

  2. 2.

If $d^{*}(S)\leqslant\delta$, then $S$ is a weak interpolation set of order
$k$ for systems of entropy $\leqslant H(\delta)+\delta\log(k-1)$.

Moreover, (1) is sharp in the following sense:

  1. 3.

There exists $S\subset\mathbb{N}$ with $d^{*}(E)=\delta$ such that $S$ is a
weak interpolation set of order $k$ for the class of systems of entropy
$\leqslant\delta\log k$.

###### Proof.

(1) Assume $d^{*}(S)>\delta$ and assume that $S$ is a weak interpolation set
of order $k$ for some system $(X,T)$. Then there exist arbitrarily large
$m\in\mathbb{N}$ and infinitely many intervals $[n+1,n+m]$ such that

$|S\cap[n+1,n+m]|\geqslant d^{*}(S)m.$

Since there are only finitely many subsets of $[1,m]$, there exists some set
$S_{m}\subset[1,m]$ for which infinitely many shifts are contained in $S$.
Construct a sequence $f:S\to\\{0,1,\ldots,k-1\\}$ which, for all $m$, contains
all possible functions on $S_{m}\to\\{0,1,\ldots,k-1\\}$ up to shift; note
that there are at least $k^{d^{*}(S)m}$ many such words. By assumption, there
exists transitive $x_{f}\in X$ and $\phi\in C(X)$ with $\phi(T^{n}x_{f})=f(n)$
for all $n\in S$. Note that the sets $S_{i}=\phi^{-1}(\\{i\\})$ are disjoint
compact sets for $0\leqslant i<k$, and so they are pairwise separated by some
distance $\varepsilon$. Then, for all $m$, by construction, there is an
$(m,\varepsilon)$-separated set of cardinality $k^{d^{*}(S)m}$, consisting of
iterates $T^{n}x_{f}$ for $n\in S$. Since $x_{f}$ is transitive, this implies
that $h(X)\geqslant d^{*}(S)>\delta$, a contradiction.

(2) Suppose $S\subset\mathbb{N}$ such that $d^{*}(S)\leqslant\delta\leqslant
1/2$ and that $f:S\rightarrow\\{0,\ldots,k-1\\}$. Define
$x_{f}:\mathbb{N}_{0}\rightarrow\\{0,\ldots,k-1\\}$ by sending all elements of
$S^{c}$ to $0$, and then define a subshift $X_{f}$ as the orbit closure of
$x_{f}$ under the left shift map.

For any $\delta^{\prime}>\delta$, by definition of $d^{*}$ there exists $M$ so
that $|S\cap I|<|I|\delta^{\prime}$ for all intervals of length at least $M$.
Therefore, for all $m\geqslant M$, $L_{m}(X_{f})\subset
S(m,\delta^{\prime},k)$, meaning that by 6.1, $h(X_{f})\leqslant
H(\delta^{\prime})+\delta^{\prime}\log(k-1)$. Since $\delta^{\prime}>\delta$
was arbitrary and $H$ is continuous, $h(X_{f})\leqslant
H(\delta)+\delta\log(k-1)$, completing the proof.

(3) The idea is to choose $S$ very regular so that even though $S$ has large
density, we can always extend a function on $S$ to a function on $\mathbb{N}$
that has small entropy.

Let $S=\\{\lfloor n/\delta\rfloor:n\in\mathbb{N}\\}$. Then $d^{*}(S)=\delta$
by definition. Define $x=\chi_{S}$; then the orbit closure $X$ of $x$ under
the left shift map is a so-called Sturmian subshift with rotation number
$\delta$. It’s known that the number of $m$-letter words in the language of
any such subshift is $m+1$ for all $m$, and it’s clear by definition that the
number of $1$s in any word of length $m$ in the language of $X$ is at most
$\lceil m\delta\rceil$ where $\lceil\cdot\rceil$ denotes the ceiling function.

Now, further define a subshift $Y$ as the set of all $0$-$1$ sequences
obtained by changing some $1$s in a point of $X$ to any letters in
$\\{0,\ldots,k-1\\}$. The number of words of any length $m$ in $Y$ is at most
$(m+1)k^{\lceil m\delta\rceil}$, so $h(Y)\leqslant\delta\log k$.

Finally, for any function $f:S\to\\{0,1,\ldots,k-1\\}$, we can extend $f$ to
$x:\mathbb{N}_{0}\to\\{0,1,\ldots,k-1\\}$ by mapping to $0$ on $S^{c}$. The
orbit closure of $f$ under the left shift is a subshift $X_{f}$ for which $x$
is a transitive point, and $X_{f}$ is contained in $Y$, so $h(X_{f})\leqslant
h(Y)\leqslant\delta\log k$. ∎

## 7\. Interpolation sets for distal systems and connections to pointwise
recurrence

A topological system $(X,T)$ on a metric space $X$ with metric $\rho$ is
called _distal_ if for all distinct $x,y\in X$ one has
$\inf_{n\in\mathbb{N}}\rho(T^{n}x,T^{n}y)>0$. The class of distal systems is
closed under taking products and inverse limits. It is a subclass of the class
of zero entropy systems and it contains all rotations on compact abelian
groups and, more generally, all nilsystems. Distal systems have been studied
in ergodic theory for a long time, partly because their measure theoretic
counterparts are core components in the structure theory of measure preserving
systems (see Furstenberg [8] and Zimmer [38, 39]).

Regarding interpolation sets for distal systems, it is shown in [29, Corollary
5.1] that any set of positive integers $S=\\{s_{1}<s_{2}<\ldots\\}$ satisfying

$\limsup_{n\in\mathbb{N}}\frac{\log(s_{n+1})}{\log(s_{n+1}-s_{n})}<\infty$
(7.1)

is an interpolation set for distal systems. (In fact, in this case, $S$ is
shown to be an interpolation set for a special subclass of distal systems
called skew products on tori.) The following natural question asked in [25] is
still open.

######  Question 7.1.

Let $S\subset\mathbb{N}$ be an interpolation set for distal systems and
$F\subset\mathbb{N}$ finite. Is it true that $S\cup F$ is an interpolation set
for distal systems?

Note that 7.1 has an affirmative answer if the class of distal systems is
replaced with rotations on compact abelian groups (cf. [24]), nilsystems (cf.
[25]), totally transitive systems (cf. 1.3), weak mixing systems (cf. 1.3),
strong mixing systems (cf. 1.3), minimal systems (cf. 1.5), or uniquely
ergodic systems (cf. 1.6). This prompts us to believe that the answer for
distal systems is also positive.

There exists a noteworthy connection between 7.1 and the notion of pointwise
recurrence in distal systems. A set $R\subset\mathbb{N}$ is a _set of
pointwise recurrence for distal systems_ if for every distal system $(X,T)$
and every point $x\in X$, we have $\inf_{n\in R}\rho(x,T^{n}x)=0$.

######  Proposition 7.2.

If every set of pointwise recurrence for distal systems can be partitioned
into two disjoint sets of pointwise recurrence for distal systems, then the
answer to 7.1 is positive.

###### Proof.

Let $S$ be an interpolation set for distal systems and let
$a\in\mathbb{N}\backslash S$. First, we show that $S-a$ is not a set of
pointwise recurrence for distal systems. Indeed, for contradiction, suppose
the opposite. Then by our hypothesis, $S-a$ can be partitioned into two sets
$A$ and $B$ both of which are sets of pointwise recurrence for distal systems.
Therefore, for any distal system $(X,T)$, any point $x\in X$,

$x\in\overline{\\{T^{n}x:n\in A\\}}\cap\overline{\\{T^{n}x:n\in B\\}}.$

(Here we use the fact that distal systems are semisimple, i.e. the orbit
closure of every point is minimal.) It follows that

$T^{a}x\in\overline{\\{T^{n}x:n\in A+a\\}}\cap\overline{\\{T^{n}x:n\in
B+a\\}}.$ (7.2)

Note that $A+a$ and $B+a$ are disjoint subsets of $S$ and so we can define a
function $f:S\to\\{0,1\\}$ with $f|_{A+a}=0$ and $f|_{B+a}=1$. Because the
intersection in (7.2) is nonempty for any distal system $(X,T)$ and any point
$x\in X$, the function $f$ cannot be extended to a sequence coming from a
distal system. This contradicts our assumption that $S$ is an interpolation
set for distal systems.

To show $S\cup\\{a\\}$ is an interpolation set for distal systems, let
$g:S\cup\\{a\\}\to[0,1]$ be an arbitrary function. Since $S$ is an
interpolation set for distal systems, there exists a distal system $(X,T)$, a
point $x_{0}\in X$ and a function $F\in C(X)$ such that $F(T^{n}x_{0})=g(n)$
for all $n\in S$. Because $S-a$ is not a set of pointwise recurrence for
distal systems, there exists a distal system $(Y,S)$ and a point $y_{0}\in Y$
such that $y_{0}\not\in\overline{\\{S^{n}y_{0}:n\in S-a\\}}$, and so
$S^{a}y_{0}\not\in\overline{\\{S^{n}y_{0}:n\in S\\}}$. Considering the product
system $(X\times Y,T\times S)$ and the point $(x_{0},y_{0})$, we have

$(T\times S)^{a}(x_{0},y_{0})\not\in\overline{\\{(T\times
S)^{n}(x_{0},y_{0}):n\in S\\}}.$

Therefore, there exists a continuous function $G:X\times Y\to[0,1]$ such that

$G(x,y)=\begin{cases}F(x),\text{ if }(x,y)\in\overline{\\{(T\times
S)^{n}(x_{0},y_{0}):n\in S\\}},\\\ g(a),\text{ if }(x,y)=(T\times
S)^{a}(x_{0},y_{0}).\end{cases}$

It follows that

$G((T\times S)^{n}(x_{0},y_{0}))=\begin{cases}F(T^{n}x_{0})=g(n),\text{ if
}n\in S,\\\ g(a),\text{ if }n=a.\end{cases}$

Since $g:S\cup\\{a\\}\to[0,1]$ is arbitrary, we conclude that $S\cup\\{a\\}$
is an interpolation set for distal systems. ∎

In light of 7.2, the following open question arises naturally and warrants
further investigation.

######  Question 7.3.

Is it true that every set of pointwise recurrence for distal systems can be
partitioned into two sets of pointwise recurrence for distal systems?

We remark that the analogues of 7.3 for compact abelian group rotations and
nilsystems were answered in [24, 25, 31].

A set $E\subset\mathbb{N}$ is an _IP-set_ if it contains an infinite sequence
and all its finite sums, that is, there are $n_{1}<n_{2}<\ldots\in\mathbb{N}$
with $\\{n_{i_{1}}+\ldots+n_{i_{k}}:k\in\mathbb{N},\leavevmode\nobreak\
i_{1}<\ldots<i_{k}\\}\subset E$. It is well known that IP-sets are sets of
pointwise recurrence in distal systems [9, Theorem 9.11]. The following
question was asked by Host, Kra, and Maass in [18] and an affirmative answer
to it would imply an affirmative answer to 7.3, because every IP-set can be
partitioned into two disjoint IP-sets.

######  Question 7.4 (Host-Kra-Maass [18, Question 3.11]).

Is it true that every set of pointwise recurrence for distal systems is an
$IP$-set?

While we don’t know how to answer 7.3, our last result shows that the stronger
7.4 has a negative answer.

######  Theorem 7.5.

There exists a set $F\subset\mathbb{N}$ that is a set of pointwise recurrence
for distal systems but for any $x,y\in F$ we have $x+y\notin F$. In
particular, $F$ is not an $IP$-set.

###### Proof.

First we show that if $F\subset\mathbb{N}$ and $F-n$ contains an $IP$-set for
all $n\in\mathbb{N}$, then $F$ is a set of pointwise recurrence for distal
systems. Indeed, let $(X,T)$ be a distal system, $U\subset X$ be open and
$x\in U$. Since the orbit closure of every point in a distal system is
minimal, there is $n\in\mathbb{N}$ such that $T^{n}x\in U$. Since $F-n$
contains an $IP$-set, by Furstenberg’s theorem [9, Theorem 9.11], there exists
$m\in F-n$ such that $T^{m}(T^{n}x)\in U$. It follows that $T^{m+n}x\in U$ and
note that $m+n\in F$.

Now we will construct the set $F$. Let $I_{1},I_{2},I_{3},\ldots$ be pairwise
disjoint infinite subsets of $\mathbb{N}$ such that $\min\\{k:k\in
I_{n}\\}\geqslant n$ for all $n$. Let $J_{n}$ be the $IP$-set generated by
$\\{10^{k}:k\in I_{n}\\}$. Then the sets $J_{1},J_{2},J_{3},\ldots$ are
pairwise disjoint and every element of $J_{n}$ is divisible by $10^{n}$.

Let $F=\bigcup_{n\in\mathbb{N}}(J_{n}+n)$. Then for every $n\in\mathbb{N}$,
the shift $F-n$ contains the $IP$-set $J_{n}$ and hence $F$ is a set of
pointwise recurrence for distal systems. It remains to show that $F$ does not
contain a triple of the form $x,y,x+y$. By contradiction, assume there are
$n,m,k$ (not necessarily distinct) such that

$\big{(}(J_{n}+n)+(J_{m}+m)\big{)}\cap(J_{k}+k)\neq\varnothing.$ (7.3)

In other words, there are $1\leqslant n_{1}<n_{2}<\ldots<n_{s}$, $1\leqslant
m_{1}<m_{2}<\ldots<m_{t}$ and $1\leqslant k_{1}<k_{2}<\ldots<k_{u}$ such that

$(10^{n_{1}}+\ldots+10^{n_{s}}+n)+(10^{m_{1}}+\ldots+10^{m_{t}}+m)=10^{k_{1}}+\ldots+10^{k_{u}}+k.$
(7.4)

By removing the terms that appear in both sides, we can assume that
$\\{k_{1},\dots,k_{u}\\}$ is disjoint from
$\\{n_{1},\dots,n_{s},m_{1},\dots,m_{t}\\}$. In particular, $k_{u}$ must be
strictly larger than both $n_{s}$ and $m_{t}$, for if, say, $n_{s}$ was the
larger, then because $k\leqslant k_{1}<10^{k_{1}}$, we would have

$10^{n_{s}}>10^{k_{1}}+\ldots+2\cdot
10^{k_{u}}>10^{k_{1}}+\ldots+10^{k_{u}}+k,$

which contradicts (7.4). That leaves the only possibility that $k_{u}$ is the
largest. But this is also impossible since

$10^{k_{u}}>2(1+10^{1}+\ldots+10^{k_{u}-1})>(10^{n_{1}}+\ldots+10^{n_{s}}+n)+(10^{m_{1}}+\ldots+10^{m_{t}}+m).$

Thus, the intersection in (7.3) is empty and hence $F$ does not contain a
triple of the form $x,y,x+y$. ∎

## 8\. Open questions

In this section, we list some natural questions that arise from our study and
may be of interest to readers.

The first question is concerned with interpolation sets for compact abelian
group rotations. A _compact abelian group rotation_ (or _group rotation_ for
short) has the form $(G,g)$ where $G$ is a compact abelian group and $g\in
G:x\mapsto g\cdot x$ for $x\in G$. It was shown in [35] that lacunary sets are
interpolation sets for group rotations. On the other hand, no subexponential
sets are interpolation sets for this class of systems [24]. For the sets $S$
that are neither lacunary nor subexponential, whether $S$ is an interpolation
set for group rotations hinges on some delicate arithmetic properties instead
of mere density. For example,
$\\{2^{n}:n\in\mathbb{N}\\}\cup\\{2^{n}+2n:n\in\mathbb{N}\\}$ is not
interpolation while
$\\{2^{n}:n\in\mathbb{N}\\}\cup\\{2^{n}+2n+1:n\in\mathbb{N}\\}$ is. Based on
these facts, it is desirable to have a complete combinatorial characterization
of interpolation sets for group rotations:

######  Problem 8.1.

Give a combinatorial characterization of interpolation sets for compact
abelian group rotations.

Group rotations is a subclass of inverse limits of nilsystems, and so our next
two questions consider interpolation sets for the latter, more general family
of systems. For $k\in\mathbb{N}$, _$k$ -step nilsystem_ is a topological
dynamical system $(X,T)$ where $X=G/\Gamma$ for $G$ a $k$-step nilpotent Lie
group and $\Gamma$ a discrete co-compact subgroup of $G$, and $T\colon X\to X$
is left translations by a fixed group element $g\in G$. Because a group
rotation is an inverse limit of $1$-step nilsytems, an interpolation set for
group rotations is interpolation for inverse limits of $k$-step nilsystems for
any $k\geqslant 1$. However, it was shown in [24] that there is an
interpolation set for inverse limits of $2$-step nilsystems which is not
interpolation for group rotations. It is unknown if the same feature continues
to hold for higher step nilsystems:

######  Question 8.2.

For $k\geqslant 2$, does there exist an interpolation set for inverse limits
of $(k+1)$-step nilsystems that is not an interpolation set for inverse limits
of $k$-step nilsystems?

Remark that 8.2 was also asked in [24, 25].

Similarly to the situation with group rotations, we have a relatively good
understanding of interpolation sets for inverse limits of nilsystems in terms
of density [2, 25, 35]: if $S$ is lacunary, $S$ is an interpolation for
inverse limits of nilsystems; if $S$ is subexponential, $S$ is not
interpolation for this class. However, this characterization does not cover
all subsets of $\mathbb{N}$ and so a complete characterization of
interpolation sets of inverse limits of nilsystems is still missing:

######  Problem 8.3.

For $k\geqslant 2$, give a characterization for interpolation sets of inverse
limits of $k$-step nilsystems.

Next, we shift our focus to interpolation sets for distal systems. The orbit
closure of every point in a distal system is minimal and therefore, by 1.5, an
interpolation set for distal systems must be non-piecewise syndetic.

Moreover it follows from (7.1) that any sequence $s_{n}$ for which the
difference between consecutive elements $s_{n+1}-s_{n}$ grows sufficiently
fast (e.g. {$\lfloor n^{1+\varepsilon}\rfloor:n\in\mathbb{N}\\}$ for any
$\varepsilon>0$), yields an interpolation set for distal systems. This
observation naturally leads to the following question, aiming to weaken the
condition (7.1):

######  Question 8.4.

Is it true that any set $S=\\{n_{1}<n_{2}<\ldots\\}$ for which the gaps
$n_{k+1}-n_{k}$ tend to infinity is an interpolation set for distal systems?

An important aspect, both in the condition (7.1) and in 8.4, is the emphasis
on the growth of gaps $n_{k+1}-n_{k}$ rather than the growth of the sequence
$n_{k}$ itself. Controlling the growth rate of $n_{k}$ is not sufficient for
interpolation in distal systems. Indeed, one can construct examples of sets of
arbitrarily fast growth rate that fail to be interpolation sets for distal
systems by considering sparse $IP$-sets. Yet $IP$-sets do not provide a
counterexample to 8.4 since they possess gaps that appear infinitely often.

We also have the following, more concrete, question about distal systems
involving prime numbers.

######  Question 8.5.

Is it true that the set of primes $\mathbb{P}$ is an interpolation set for
distal systems?

According to the celebrated Zhang’s theorem [37], the sequence of gaps between
consecutive primes does not go to infinity. Therefore, 8.5 is not a special
case of 8.4. In order for $\mathbb{P}$ to be an interpolation set for distal
systems, for any partition $\mathbb{P}=A\cup B$, there must be a distal system
$(X,T)$ and a point $x\in X$ such that the closures $\overline{\\{T^{p}x:p\in
A\\}}$ and $\overline{\\{T^{p}x:p\in B\\}}$ are disjoint. Related to this,
recently, Kanigowski, Lemańczyk, and Radziwiłł [20] proved that the prime
number theorem holds for skew products on $\mathbb{T}^{2}$ with
$T(x,y)=(x+\alpha,y+h(x))$ where $h$ is an analytic function. More precisely,
they showed that for these systems $(X,T)$ and for any $x\in X$, the set
$\\{T^{p}x:p\in\mathbb{P}\\}$ is uniformly distributed on $\mathbb{T}^{2}$.
That means for any partition $\mathbb{P}=A\cup B$,

$\overline{\\{T^{p}x:p\in A\\}}\cup\overline{\\{T^{p}:p\in
B\\}}=\mathbb{T}^{2}$

and so these two closures cannot be disjoint. As a result, $\mathbb{P}$ is not
interpolation for the skew products on $\mathbb{T}^{2}$ with analytic skewing
functions. That being said, since the class of distal systems is much larger
than the aforementioned skew products, the answer to 8.5 may be positive.

The next question is concerned with the threshold for density of weak
interpolation sets for uniquely ergodic systems. It follows from the proof of
[36, Theorem 8.1] that for any $\varepsilon>0$, there exists a set $S$ which
is a weak interpolation set of order $2$ for uniquely ergodic systems such
that $d^{*}(S)>1/2-\varepsilon$. The proof can be generalized to arbitrary
$k\in\mathbb{N}$ to show that there exists a set $S$ which is a weak
interpolation set of order $k$ for uniquely ergodic systems such that
$d^{*}(S)>1/k-\varepsilon$. However, from the proof of 1.6, there is no such a
set that satisfies $d^{*}(S)>1/k$. These facts lead to the following question.

######  Question 8.6.

For $k\in\mathbb{N}$, is there a set $S$ which is a weak interpolation set of
order $k$ for uniquely ergodic systems such that $d^{*}(S)=1/k$?

Among the classes of systems considered in this paper, for minimal systems,
totally minimal systems, systems of zero entropy, and systems of finite
entropy, weak interpolation of order $2$ is equivalent to weak interpolation
of all orders. On the other hand, for uniquely ergodic systems and strictly
ergodic systems, these two notions are not equivalent. Our next question
explores this equivalence for the class of totally transitive systems. More
precisely, 1.3 implies that a weak interpolation set of all orders for totally
transitive systems cannot be syndetic. Nevertheless, we do not know whether a
weak interpolation set of order $2$ for this class can be syndetic or not.

######  Question 8.7.

Does there exist a syndetic set which is weak interpolation of order $2$ for
the class of totally transitive systems?

Our last question is concerned with the necessity of the hypothesis of 1.7.
Recall that 1.7 shows that a syndetic set $S$ satisfying the orbit closure of
$1_{S}$ under the left shift has zero entropy cannot be weak interpolation of
order $2$ for uniquely ergodic systems. We do not know whether the entropy
condition is necessary and so the following question is open:

######  Question 8.8.

Does there exist a syndetic set which is weak interpolation of order $2$ for
the class of uniquely ergodic systems?

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Andreas Koutsogiannis
Aristotle University of Thessaloniki

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Anh N. Le
University of Denver

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Joel Moreira
University of Warwick

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Ronnie Pavlov
University of Denver

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Florian K. Richter
École Polytechnique Fédérale de Lausanne (EPFL)

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