Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems

We show that an $R^d$-topological dynamical system equipped with an invariant ergodic measure has discrete spectrum if and only it is $\mu$-mean equicontinuous (proven for $Z^d$ before). In order to do this we introduce mean equicontinuity and mean sensitivity with respect to a function. We study this notion in the topological and measure theoretic setting. In the measure theoretic case we characterize almost periodic functions and in the topological case we show that weakly almost periodic functions are mean equicontinuous (the converse does not hold).

systems are the periodic systems, i.e. when for every x ∈ X there exists n x > 0 such that T nx x = x; followed by the equicontinuous systems i.e. when for every ε > 0 there exists δ > 0 such that if d(x, y) ≤ δ then d(T i x, T i y) ≤ ε for every i ≥ 0. This notion means that the system if highly predictable in the following sense: if you only know x is inside a small δ−neighbourhood you will be able to predict with ε−precision the orbit of x. Several weaker notions like distality, nullness, tameness, mean equicontinuity, among others have been studied. In particular we are interested in mean equicontinuity. A system is mean equicontinuous , if for every ε > 0 there exists δ > 0 such that if d(x, y) ≤ δ then lim sup 1 n n i=1 d(T i x, T i y) ≤ ε . Here we have predictability not for every i but on sets of high density (see Proposition 1.23). According to Asulander [2], Fomin introduced these systems in [11].
Mean equicontinuity has received interest in recent years due to connections with ergodic properties of measurable dynamical systems, i.e. dynamical systems equipped with an invariant probability measure. In particular, it has been shown that using a measure theoretic version of mean equicontinuity one can characterize when a measure-preserving system has discrete spectrum [14] (i.e. when the eigenfunctions of the Koopman operator generate L 2 ) and when the maximal equicontinuous factor is actually an isomorphism (i.e. when the continuous eigenfunctions of the Koopman operator generate L 2 ) [22,8] . The proof of the characterization of discrete spectrum using mean equicontinuity in [14] does not hold for continuous actions (e.g. R d ). In this paper we prove that the characterization of discrete spectrum also holds for abelian G-actions (Theorem 3.10). To do this we introduce mean equicontinuity with respect to a function. We say that (X, T ) is mean equicontinuous with respect to f : X → C if for every ε > 0 there exists δ > 0 such that if d(x, y) ≤ δ then lim sup 1 n n i=1 f (T i x) − f (T i y) ≤ ε. Actually we will see that a system is mean equicontinuous if and only if it is mean equicontinuous with respect to every continuous function. Using the measure theoretic version of this concept we can characterize almost periodic functions (Corollary 3.8). Almost periodic functions are classical objects in ergodic theory that are used to characterize discrete spectrum and weak mixing.
Another important concept that appears in the paper is sensitivity. A topological dynamical system is sensitive if there exists ε > 0 such that for every x, y ∈ X there exists i such that d(T i x, T i y) ≥ ε and mean sensitive if there exists ε > 0 such that for every x, y ∈ X we have lim sup 1 We study similar concepts in three different categories of dynamical systems (measurable, topological, and measurable and topological) which divide sections of the paper. The paper is arranged as follows.
In Section 1 we consider measure preserving systems (MPS) without any topology. The main objective of this section is to characterize almost periodic functions. To do this we introduce a purely measure theoretic notion (no topology) of mean sensitivity with respect to an L 2 function f (µ − f −mean sensitivity see Definition 1.12). We then characterize this concept as the negation of almost periodicity of f (when the action is Abelian). As a corollary, we characterize systems with discrete spectrum as those ergodic systems that are not mean sensitive with respect to all L 2 functions; and we characterize weakly mixing systems as those ergodic systems which are mean sensitive with respect to all non-constant L 2 functions. Note that while it is possible to develop a purely measure theoric version of mean sensitivity, we did not find a natural purely measure theoretic notion of mean equicontinuity (other than negation of mean sensitivity).
In Section 2 we consider topological dynamical systems (TDS) (without measures). We define the purely topological versions of mean equicontinuity and mean sensitivity of a TDS relative to a given continuous function f . We show that in the case of a minimal TDS, mean sensitivity and mean equicontinuity (relative to f ) are complementary notions and partition C(X). Finally, for a TDS we give results on the relationships between topological almost periodicity defined by Ellis in [10], and mean equicontinuity. In particular we show that if a function is weakly almost periodic then the system is mean equicontinuous with respect to the function (the converse does not hold).
In Section 3 we consider topological dynamical systems equipped with invariant probability measures. It is in this category where we can define a hybrid (measure theoretic and topological) form of mean equicontinuity relative to a given function f and a measure µ (µ − f −mean equicontinuity). First we show that this notion is complementary to the measure-theoretic notion of mean sensitivity relative to f (defined in Section 1). Finally we present a characterization of discrete spectrum for TDS equipped with ergodic measures (Theorem 3.10). In Section 4 we apply this result to characterize quasicrystalline behaviour on Delone sets. Finally we compare our results with other characterizations of discrete spectrum in terms of topological averages (e.g. [17] [21]).
In this paper G denotes an abelian locally compact group (with operation + and identity e). Every locally compact group has a unique measure that is invariant under rotations, known as Haar measure on G. We will denote this measure with ν. In particular if G = R d , then ν is the Lebesgue measure, and if G countable ν is the cardinality.
We say a group G is amenable if it contains a Følner sequence. Every Abelian group is amenable. In particular if G = Z d or G = R d then F n := [−n, n] d is a Følner sequence.
Definition 0.1 Let S ⊂ G and {F n } n∈N a Følner sequence. We define the lower density of S by and the upper density of S by

Measure theoretical results
A G− measure preserving system (G−MPS) is a quadruple (X, Σ, µ, T ) where (X, Σ, µ) is a standard probability space and is a group action (that is T e x = x and T g T j x = T g+j x ∀x ∈ X) and T j : X → X is measure preserving on (X, Σ, µ) for every j ∈ G. When it is not needed we will omit writing Σ. A Z-measure preserving system is generated by a measure preserving invertible transformation T : (X, µ) → (X, µ) on a Lebesgue probability space (X, µ).
We say a measurable subset is invariant if it is invariant under every T i ; we say a G−MPS is an ergodic if every measurable invariant set has measure 0 or 1. The collection of measurable sets with positive µ-measure is denoted Σ + . A measure preserving system (X, µ, T ) generates a family of unitary linear operators (known as the Koopman operators) on the complex Hilbert space We denote the inner product of L 2 (X, µ) by ·, · . Definition 1.1 Let (X, µ, T ) be an ergodic G−MPS and {F n } a Følner sequence. Let A ⊂ X be a measurable set. We say x ∈ X is a generic point for A if Let f be a measurable function. We say x ∈ X is a generic point for f if Lindenstrauss proved that for every amenable group there exists a Følner sequence {F n } that satisfies the pointwise ergodic theorem i.e. that for every f ∈ L 1 (X, µ) almost every x ∈ X is a generic point for f [23]. Remark 1.2 From now on we associate to every G a Følner sequence {F n } that satisfies the pointwise ergodic theorem.

Almost periodic and µ−mean sensitive functions
Let S 1 ⊂ C be the unit circle. Given G we denote with G the (Pontryagin) dual group, which is composed by the continuous group homomorphisms from G to S 1 . Given g ∈ G and w ∈ G, we define (w, j) := w(j). Definition 1.3 Let (X, µ, T ) be an ergodic G−MPS and f ∈ L 2 (X, µ) (complexvalued functions).
·We say f is an almost periodic function if cl U j f : j ∈ G is compact as a subset of L 2 (X, µ). We denote the set of almost periodic functions by H ap .
·We say f = 0 is an eigenfunction of (X, µ, T ) if there exists w∈ G such that The following results are well known (see for example [12][13]). Proposition 1.6 Let (X, µ, T ) be a G−MPS.
1. H ap is the closure of the set spanned by the eigenfunctions of T.
2. f ∈ L 2 (X, µ) is almost periodic if and only if for every ε > 0 there exists a syndetic set S ⊂ G such that f − U j f 2 dµ ≤ ε for every j ∈ S.
3. The product of two almost periodic functions is almost periodic.
4. If (X, µ, T ) is ergodic and f is an eigenfunction then |f | is constant almost everywhere.
Definition 1.7 Let (X, µ, T ) and (X ′ , µ ′ , T ′ ) be two G−MPS. We say they are isomorphic if there exists an a.e. bijective and measure preserving function ·(X, µ, T ) has discrete spectrum if there exists an orthonormal basis for L 2 (X, µ) which consists of eigenfunctions of (X, µ, T ).
·(X, µ, T ) has continuous spectrum if the only eigenfunctions are the constant functions.
The following two results are due to Halmos and Von-Neumann (see for example [27]: note that the original proof is stated for G = Z, nonetheless the same proof is valid for abelian group actions). Theorem 1.9 Let (X, µ, T ) be an ergodic system. The following are equivalent: ·(X, µ, T ) has discrete spectrum. ·(X, µ, T ) is a measurable isometry. ·L 2 (X, µ) = H ap Theorem 1.10 Let (X, µ, T ) be an ergodic system. The following are equivalent: ·(X, µ, T ) has continuous spectrum. ·(X, µ, T ) is weakly mixing. ·H ap consists only of constant functions.
Let F denote the set of all functions α : We define the following pseudometric on F : It is not difficult to show that this is indeed a pseudometric (i.e. satisfies symmetry and the triangle inequality). For any f ∈ L 2 (X, µ) we introduce the following pseudometric on a set of full measure in X.
Using Cauchy-Schwartz and the pointwise ergodic theorem on can check that d f (x, y) is finite-valued for all x, y ∈ X that are generic points for f 2 . Letting α(j) = f (T j x), β(j) = f (T j y), we see that d f (x, y) = e(α, β) and thus it is a pseudo-metric on the set of generic points for f 2 . Whenever we use the pseudometric d f we only consider the points that are generic for f 2 .
In this case we also say that f is µ−mean sensitive and ε is a µ-sensitivity constant for f . We denote the set of µ−mean sensitive functions by H ms .
Remark 1. 13 We will see in Section 1.3 that for f ∈ L ∞ (X, µ), there are equivalent pseudometrics that give different perspectives on mean sensitivity.
Proof. Let f ∈ H ms and let ε > 0 be a mean sensitivity constant for f . Let g ∈ L 2 (X, µ) such that |f − g| 2 dµ ≤ (ε/4). By the pointwise ergodic theorem there exists Y ⊂ X with µ(Y ) = 1 and Since e(α, β) is a pseudometric we obtain Thus g ∈ H ms and so H ms ⊂ L 2 (X, µ) is open.
Proof. We first show that H c ap ⊆ H ms . Assume that f ∈ H c ap , and so cl U j f : j ∈ G is not compact. This implies it is not totally bounded, hence there exists ε > 0 and an infinite subset S ⊂ G such that By the pointwise ergodic theorem, there Let p := T s z ∈ A and q := T t z ∈ A. Since G is commutative we have that We conclude that for every A ∈ Σ + there exist p, q ∈ A such that Now we show that H ap ⊆ H c ms . Let ε > 0 and g be a measurable function. There exists a set X ε ∈ Σ + such that |g(x) − g(y)| 2 ≤ ε for every x, y ∈ X ε . Now assume that g is an eigenfunction and so there exists w∈R d such that U j (g) = e 2πi w,j g. Then for every x, y ∈ X ε and every j ∈ G we have that Thus g ∈ H c ms . Using Lemma 1.14 and Proposition 1.6 (part 1), we get that H ap ⊂ H c ms , as desired.
Note that the proof above shows that in the definition of H ms we can replace lim sup with lim, i.e., f ∈ H ms if and only if there exists ε > 0 such that for every A ∈ Σ + there exists x, y ∈ A such that Proof. 1) Follows from Theorems 1.9 and 1.15.
The proof of Theorem 1.20 is analogous to the proof of Theorem 2.6 of [14] but uses d f instead of the Besicovitch pseudometric. The proof is short so we write it for completeness.
We say that ε is a µ-mean expansive constant for f .
Since g is T -invariant we conclude that g(x) is constant for almost every x ∈ X and equal to µ × µ {(x, y) : d f (x, y) ≤ ε} .
is µ − f −mean sensitive (with µ − f −mean sensitivity constant ε) and that 3) is not satisfied. This means there exists x ∈ X such that µ(B f ε/2 (x)) > 0. For any y, z ∈ B f ε/2 (x), we have that d f (y, z) ≤ ε. This contradicts the assumption that (X, T ) is µ − f −mean sensitive.

Equivalent pseudometrics
We consider the following pseudometrics, which turn out to be equivalent to d f when f is bounded.
, and ρ f (x, y) are equivalent, i.e., generate the same topology.
Proof. Without loss of generality we may assume that |f | Then for all δ < ε, we obtain yields a contradiction. We conclude that ρ f (x, y) < ε.
This implies we can use d ′ f (x, y) or ρ f (x, y) as alternatives to d f (x, y) in the definition of µ-f -mean sensitivity.

Topological results
A G−topological dynamical system (TDS) is a pair (X, T ), where X is a compact metric space and T is a group action with T j : X → X continuous for every j ∈ G.
The closed ε-balls of X will be denoted by B ε (x), and the collection of Borel sets of X by B X .

Mean equicontinuity and mean sensitivity
Using subadditivity of lim sup, it is not hard to show that d b and ρ b are indeed pseudometrics. The subscript "b" stands for "Besicovitch," as these are versions of the Besicovitch pseudometric.  We say x ∈ X is a mean equicontinuity point if for every ε > 0 there exists δ > 0 such that if y ∈ B δ (x) then d b (x, y) < ε.
We say (X, T ) is mean equicontinuous (or mean-L-stable) if every x ∈ X is a mean equicontinuity point. We say (X, T ) is almost mean equicontinuous if the set of mean equicontinuity points is residual.
Using Lemma 2.2, we conclude that (X, T ) is mean sensitive if and only if for every non-empty open set U there exist x, y ∈ U such that ρ b (x, y) > ε; and x ∈ X is a mean equicontinuity point if and only if for every ε > 0 there exists δ > 0 such that if y ∈ B δ (x) then ρ b (x, y) < ε.

Remark 2.5
If (X, T ) is mean equicontinuous, then it is uniformly mean equicontinuous in the sense that for every ε > 0 there exists δ > 0 such that if d(x, y) ≤ δ then D(i ∈ G : d(T i x, T i y) > ε) < ε. One way to see this is by identifying points x, y such that d b (x, y) = 0. Namely, we obtain a metric space (X/d b , d b ) and a natural projection π b : (X, d) → (X/d b , d b ) by mapping x ∈ X to its equivalence class in X/d b . Then mean equicontinuity of (X, T ) means that π b is continuous and hence uniformly continuous.
In [1][3] dichotomies between equicontinuity and sensitivity were exhibited. In [14] (and independently in [22]) the following dichotomies involving mean sensitivity and mean equicontinuity were proved. Definition 2.6 Let (X, T ) be a G−TDS. We say x ∈ X is a transitive point if T i x : i ∈ G is dense. We say (X, T ) is transitive if X contains a residual set of transitive points. If every x ∈ X is transitive then we say the system is minimal.
Theorem 2.7 [14][22]A transitive system is either almost mean equicontinuous or mean sensitive. A minimal system is either mean equicontinuous or mean sensitive.
In the next subsection we will introduce versions of mean equicontinuity and mean sensitivity relative to given f ∈ C(X) and we will establish similar dichotomies. In this case we also say that f is mean sensitive for (X, T ), and we denote the set of all mean sensitive functions for (X, T ) by C ms .

f −Mean equicontinuity and f −Mean sensitivity
The notion of f −mean equicontinuity appeared in [26]. Definition 2.9 Let (X, T ) be a G−TDS and f ∈ C(X). We say x ∈ X is a f −mean equicontinuity point if for every ε > 0 there exists δ > 0 such that if d(x, y) ≤ δ then d f (x, y) ≤ ε. We say (X, T ) is f −mean equicontinuous if all x ∈ X are f −mean equicontinuous. In this case we also say that f is mean equicontinuous function for (X, T ); we denote the set of mean equicontinuous functions by C me . We say (X, T ) is f − almost mean equicontinuous if the set of mean equicontinuity points is residual.

Remark 2.10
Since any continuous function on X is bounded, we can use Proposition 1.23, to characterize f −mean equicontinuity and f −mean sensitivity by replacing d f (x, y) with ρ f (x, y) in the definition. We will mainly use the latter.
Again using the idea that a continuous function on a compact metric set is uniformly continuous, it is not hard to see that if (X, T ) is f −mean equicontinuous, then it is uniformly f −mean equicontinuous in the sense that for every ε > 0 there exists δ > 0 such that if d(x, y) ≤ δ then d f (x, y) ≤ ε.

Definition 2.11
Let f ∈ C(X). We denote the set of f −mean equicontinuity points by E f and we define Note that Proof. Let j ∈ G, ε > 0, and x ∈ T −j E f ε . There exists η > 0 such that if d(T j x, z) ≤ η and d(T j x, y) ≤ η then There also exists δ > 0 such that if d(x, y) ≤ δ then d( and so E f ε is inversely invariant. It follows that E f is also inversely invariant. Let x ∈ E f ε and δ > 0 satisfy the defining property of E f ε . If d(x, w) < δ/2 then w ∈ E f ε ; indeed if y, z ∈ B δ/2 (w) then y, z ∈ B δ (x). So, E f ε is open. It is not hard to see that f −mean sensitive systems have no f −mean equicontinuity points. The proof of the following dichotomies is very similar to the proof of Theorem 2.7; we include it for completeness. Theorem 2. 13 Let f ∈ C(X). A transitive system is either f −almost mean equicontinuous or f −mean sensitive. A minimal system is either f −mean equicontinuous or f −mean sensitive.
Proof. First, we show that if (X, T ) is a transitive system then for every ε, E f ε is either empty or dense. Assume E f ε is non-empty and not dense. Then U = X (cl(E f ε )) is a non-empty open set. Since the system is transitive and E f ε is nonempty and open (by Lemma 2.12) there exists t ∈ G such that U ∩T −t (E f ε ) is non empty. By Lemma 2.12 we have that If E f ε is non-empty for every ε > 0 then we have that E f = ∩ n≥1 E f 1/n is a residual set; hence the system is f -almost mean equicontinuous.
If there exists ε > 0 such that E f ε is empty, then for any open set U there exist y, z ∈ U such that D i ∈ G : Now suppose (X, T ) is minimal and f -almost mean equicontinuous. For every x ∈ X and every ε > 0 there exists x ∈ E f . In the following result we use a technique from the topological Halmos Von-Neumann Theorem (e.g. see Chapter 5.5 [27]).
Theorem 2.14 Let (X, T ) be a G−TDS. Then (X, T ) is mean equicontinuous if and only if it is f −mean equicontinuous for every f ∈ C(X) (i.e. C(X) = C me ).

This implies that
Hence (X, T ) is f −mean equicontinuous. ⇐ Let {f n } be a collection of functions such that |f n | ≤ 1 and the closure of its linear span is C(X).
Such a collection separates points of X, and so is a metric on X. Let ε > 0 and choose N so that ∞ n=N +1 2/2 n ≤ ε/2. There exists δ > 0 such that if d(x, y) ≤ δ then So, the identity map from (X, d) to (X, d) is a bijective continuous map on a compact metric space and hence is a homeomorphism. We conclude that d and d are equivalent metrics. It is not hard to see that mean equicontinuity is invariant under a change of equivalent metrics. We will show (X, T ) is mean equicontinuous with respect to d.
Let ε > 0 and choose N so that for every 1 ≤ m ≤ N. Since d and d are equivalent metrics, there exists η > 0 such that if d(x, y) ≤ η, then d(x, y) ≤ δ, and so lim sup So, (X, T ) is mean equicontinuous with respect to d.

Weakly almost periodic functions
In [10], Ellis defined two topological notions of almost periodic functions in C(X). We consider C(X) as a Banach space with the ℓ ∞ norm. Two classic topologies studied on Banach spaces are: the strong topology (given by the norm) and the weak topology (the coarsest topology such that each element in the dual space is continuous). Since X is compact f n → f in the weak topology if and only if f n (x) → f (x) for every x ∈ X. Definition 2.15 Let (X, T ) be a G−TDS. We say f ∈ C(X) is almost periodic if U j f : j ∈ G ⊂ C(X) is precompact with respect to the strong topology. We say f ∈ C(X) is weakly almost periodic (WAP) if U j f : j ∈ G ⊂ C(X) is precompact with respect to the weak topology. If every f ∈ C(X) is WAP then we say (X, T ) is weakly almost periodic.
A system is equicontinuous if and only if every function is almost periodic. A minimal system is equicontinuous if and only if it is weakly almost periodic [10]; nonetheless there are examples of transitive weakly almost periodic systems that are not equicontinuous. We will show these systems are always mean equicontinuous.
Lemma 2. 16 Let (X, T ) be a G−TDS. If f is almost periodic then it is mean equicontinuous.
Proof. Let ε > 0. There exists a finite set F ⊂ G such that for every j ∈ G there exists i j ∈ F such that U j f − U ij f ∞ ≤ ε (where U is the Koopman operator on C(X)). By uniform continuity there exists δ > 0 such that if d(x, y) ≤ δ then and thus By the equivalence of the metrics, d f (x, y) and d ′ f (x, y), we conclude that f is mean equicontinuous.
Proposition 2.17 Let G be a countable group and (X, T ) be a uniquely ergodic G−TDS. If f is weakly almost periodic then it is mean equicontinuous.
Proof. Let f ∈ C(X) be weakly almost periodic. As a consequence of the de Leeuw-Glicksberg decomposition (see Theorem 1.51 in [16]) we have that where g is almost periodic and lim 1 |Fn| i∈Fn h(T i x) = 0 for every x. This implies h is mean equicontinuous. By the previous lemma g is mean equicontinuous; we conclude f is mean equicontinuous.
Using that transitive WAP systems are uniquely ergodic (Lemma 1.50 in [16]), the previous Proposition and Theorem 2.14 we obtain the following result.

Measure theoretic and topological results
In this section we present hybrid results, that is, results that reflect the topology as well as the measure theoretic structure of the sytem.
We remind the reader that given a metric space X and a Borel probability measure µ we denote the Borel sets with B X and the Borel sets with positive measure with B + X . We say (X, µ, T ) is an ergodic G−TDS if (X, T ) is a G−TDS, µ a Borel probability measure and (X, µ, T ) an ergodic G−MPS.

µ−Mean equicontinuity
Definition 3.1 Let (X, T ) be a G−TDS, µ an invariant Borel probability measure. We say (X, T ) is µ−mean equicontinuous if for every τ > 0 there exists a compact set M ⊂ X, with µ(M ) ≥ 1 − τ, such that for every ε > 0 there exists δ > 0 such that whenever x, y ∈ M and d(x, y) ≤ δ then Remark 3.2 It is not hard to see that d b (·, ·) is a Borel function.
Also, denoting the closed ε−balls of the pseudometric d b by B b ε (x), for every ε > 0 and every Measure theoretic forms of sensitivity for G−TDS have been studied in [15], [7], and [18]. In particular in [18] it was shown that ergodic G−TDS are either µ−equicontinuous or µ−sensitive. In [14] µ−mean sensitivity was introduced.
In the next subsection we will obtain a similar result for µ − f −mean sensitivity.

µ − f −Mean equicontinuity
For the definition of d f and B f ε see Definition 1.12 and Definition 1.19.
Definition 3.5 Let (X, T ) be a G−TDS, µ a Borel probability measure and f ∈ L 2 (X, µ). We say (X, T ) is µ − f −mean equicontinuous if for every τ > 0 there exists a compact set M ⊂ X, with µ(M ) ≥ 1 − τ, such that for every ε > 0 there exists δ > 0 such that whenever x, y ∈ M and d(x, y) ≤ δ then In this case we say f is µ−mean equicontinuous. We denote the set of µ−mean equicontinuous functions by H me .
The following fact is well known (for a proof see [14]).
Given by Theorem 1.20.
The following proofs are similar to the proof of Theorem 3.4.
2) ⇒ 4) If (X, T ) is µ-f -mean-expansive, then there is a set Z ⊂ X × X s.t µ(Z) = 1 and for all (x, y) ∈ Z, we have d f (x, y) > ε. Suppose that (X, T ) is also µf -mean equicontinuous. Then there is a compact set M ⊆ X with positive measure and δ > 0 such that if x, y ∈ M and d(x, y) ≤ δ, then d f (x, y) ≤ ε. Since M is compact, it can be covered by finitely many δ/2-balls. For at least one of these balls B, we have µ(B ∩ M ) > 0 and so (( For any (x, y) in this intersection, we have d f (x, y) ≤ ε and d f (x, y) > ε, a contradiction.

4) ⇒ 3)
Suppose 3) is not satisfied. By Lemma 1.18 we have that for every n ∈ N there exists a set of full measure Y n such that is not separable then by Lemma 3.6 there exists ε > 0 and an uncountable set A such that for every x, y ∈ A with x = y we have that B f ε (x) ∩ B f ε (y) = ∅. For any 1/n < ε, this is a contradiction to the fact that µ(B f ε (x)) > 0 for all x ∈ Y n . Hence, (Y d f , d f ) is separable. Using Lusin's Theorem we conclude (X, T ) is µ−f −mean equicontinuous (this is done exactly as in Proposition 19 in [14]).

Relationship to results of quasicrystals
Of particular interest in the mathematical theory of quasicrystals and aperiodic order is studying long range order that Delone sets may exhibit; Delone sets are uniformly discrete and relatively dense subsets of R d (see [4] and [19] for recent general expositions that contain the main definitions in this section). A Delone set is crystalline if it is periodic (in all d directions). These are the most ordered Delone sets. We say a Delone set is quasicrystalline if it has pure point diffraction spectrum (this notion is defined using Fourier transforms) and uniform patch frequency (configurations of patches of points appear with a uniform frequency). Periodic Delone sets always have pure point diffraction spectrum and uniform patch frequency. Initially it was not known if non-periodic subsets could have pure point diffraction spectrum; nowadays there are many examples, perhaps the best known is the Delone set associated to the Penrose tiling [25]. To every Delone set we can associate an R d −topological dynamical system ( uniquely ergodic when the set has uniform patch frequency), where the R d −action is defined by the shifts in the d directions. In [6] Baake, Lenz and Moody showed that a Delone set is crystalline if and only if the associated R dtopological dynamical system is equicontinuous (this means that the family of the shifts is equicontinuous). This motivates the following question; can quasicrystals be characterized with a weaker forms of equicontinuity? Combining the work of several papers it is known that a Delone set is quasicrystalline if and only if the associated dynamical system is uniquely ergodic and has discrete spectrum [9,20,5]. Using this and Theorem 3.10 we obtain that a Delone set is quasicrystalline if and only if the associated dynamical system is uniquely ergodic and µ-mean equicontinuous (with µ the unique invariant measure).
There are other characterizations that use the Besicovitch pseudometric to characterize discrete spectrum. In [17] it is proven that a (minimal) uniquely ergodic Delone dynamical system has discrete spectrum if and only if it is mean almost periodic. That is, for every x ∈ X and ε > 0 there exists a syndetic set S ⊂ R d , such that d b (T i x, x) ≤ ε for every i ∈ S. This result is different from ours in the following senses; besides the clear distinction that mean almost periodicity and mean equicontinuity are different concepts we see that this result applies only for minimal systems and the condition mean almost periodicity does not depend on the measure. Our characterization using µ−mean equicontinuity requires the property of mean equicontinuity only on sets with large measure. Nonetheless there is a relationship. It is not hard to see that every minimal mean equicontinuous G−TDS is mean almost periodic. Let ε > 0 and δ > 0 the number such that if d(x, y) ≤ δ then d b (x, y) ≤ ε. Since (X, T ) is minimal every point is almost periodic, thus there exists a syndetic set S ⊂ R d , such that d(T i x, x) ≤ δ for every i ∈ S; this implies d b (T i x, x) ≤ ε for every i ∈ S. On [21] there is a similar characterization that uses Bohr almost periodicity and applies to general G−TDS.