Escaping orbits are also rare in the almost periodic Fermi-Ulam ping-pong

We study the one-dimensional Fermi-Ulam ping-pong problem with a Bohr almost periodic forcing function and show that the set of initial condition leading to escaping orbits typically has Lebesgue measure zero.


Introduction
The Fermi-Ulam ping-pong is a model describing how charged particles bounce off magnetic mirrors and thus gain energy. They undergo the so called Fermi acceleration and one central question is whether the particles velocities can get close to the speed of light that way. The model was introduced by Fermi [Fer49] in order to explain the origin of high energy cosmic radiation. A common one-dimensional mathematical formulation of this problem is as follows: The point particle bounces completely elastically between two vertical plates of infinite mass, one fixed at x = 0 and one moving in time as x = p(t) for some forcing function p = p(t) > 0. The particle alternately hits the walls and experiences no external force in between the collisions. The motion can be described by the successor map f : (t 0 , v 0 ) → (t 1 , v 1 ), mapping the time t 0 ∈ R of an impact at the left plate x = 0 and the corresponding velocity v 0 > 0 right after the collision to (t 1 , v 1 ), representing the subsequent impact at x = 0. Since one is interested in the long term behavior, we study the forward iterates (t n , v n ) = f n (t 0 , v 0 ) for n ∈ N and in particular the 'escaping set' consisting of initial data, which lead to infinitely fast particles. The most studied case is that of a periodic forcing p(t). Ulam [Ula61] conjectured an increase in energy with time on the average. Based on some numerical simulations, he however realized that rather large fluctuations and no clear gain in energy seemed to be the typical behavior.
Two decades later, the development of KAM theory allowed to prove that the conjecture is indeed false. If the forcing p is sufficiently smooth, all orbits stay bounded in the phase space, since the existence of invariant curves prevents the orbits from escaping [LL91,Pus83]. The proofs are based on Moser's twist thoerem [Mos62], which relies on the higher regularity. And indeed, Zharnitsky [Zha98] showed the existence of escaping orbits if only continuity is imposed on p. In the non-periodic case, one can even find C ∞forcings with this behavior [KO10]. More recently, Dolgopyat and De Simoi developed a new approach. They consider the periodic case and study some maps which are basically approximations of the successor map f . This way they could prove several results regarding the Lebesgue measure of the escaping set E [Dol08b,Dol08a,dSD12,Sim13]. Finally, Zharnitsky [Zha00] investigated the case of a quasi-periodic forcing function whose frequencies satisfy a Diophantine inequality. Again, using an invariant curve theorem, he was able to show that the velocity of every particle is uniformly bounded in time. Since no such theorem is available if the Diophantine condition is dropped, a different approach is necessary in this case. This was done by Kunze and Ortega in [KO18]. They apply a refined version of the Poincaré recurrence theorem due to Dolgopyat [Dol] to the set of initial condition leading to unbounded orbits, and thereby show that most orbits are recurrent. Thus, typically the escaping set E will have Lebesgue measure zero. Now, in this work we will give an affirmative answer to the question raised in [KO18] whether this result can be generalized to the almost periodic case. Indeed, most of their arguments translate naturally into the language of Bohr almost periodic functions. Our main theorem (Theorem 5.1) states that the escaping set E is most likely to have measure zero, provided the almost periodic forcing p is sufficiently smooth. In order to explain more precisely what we mean by 'most likely', we first need to introduce some properties and notation regarding almost periodic functions. This is done in section 2. Subsequently we will study measure-preserving successor maps of a certain type and their iterations. We end this part by stating Theorem 3.1, a slightly generalized version of a theorem by Kunze and Ortega [KO18], which describes conditions under which the escaping set typically will have measure zero. This will be the most important tool and its proof will be given in the following section. Then, in the last section we discuss the ping-pong model in more detail and finally state and prove the main theorem.
2 Almost periodic functions and their representation

Compact topological groups and minimal flows
Let Ω be a commutative topological group, which is metrizable and compact. We will consider the group operation to be additive. Moreover, suppose there is a continuous homomorphism ψ : R → Ω, such that the image ψ(R) is dense in Ω. This function ψ induces a canonical flow on Ω, namely This flow is minimal, since ω · R = ω + ψ(R) = ω + ψ(R) = Ω holds for every ω ∈ Ω. Let us also note that in general ψ can be nontrivial and periodic, but this happens if and only if Ω S 1 [OT06]. Now consider the unit circle S 1 = {z ∈ C : |z| = 1} and a continuous homomorphism ϕ : Ω → S 1 . Such functions ϕ are called characters and together with the point wise product they form a group, the so called dual group Ω * . Its trivial element is the constant map with value 1. It is a well known fact that nontrivial characters exist, whenever Ω is nontrivial [Pon66]. Also non-compact groups admit a dual group. Crucial to us will be the fact that Now, for a nontrivial character ϕ ∈ Ω * we define Then Σ is a compact subgroup of Ω. If in addition Ω ≇ S 1 , it can be shown that Σ is perfect [OT06]. This subgroup will act as a global cross section to the flow on Ω. Concerning this, note that since ϕ • ψ describes a nontrivial character of R, there is a unique α 0 such that ϕ(ψ(t)) = e iαt for all t ∈ R. Therefore, the minimal period of this function, can be seen as a returning time on Σ in the following sense. If we denote by τ(ω) the unique number in [0, S ) such that ϕ(ω) = e iατ(ω) , then one has ϕ(ω · t) = ϕ(ω + ψ(t)) = ϕ(ω)ϕ(ψ(t)) = e iατ(ω) e iαt and thus ω · t ∈ Σ ⇔ t ∈ −τ(ω) + S Z.
Example 2.1. One important example for such a group Ω is the N-Torus T N , where T = R/Z. We will denote classes in T N byθ = θ + Z. Then, the image of the homomorphism winds densely around the torus T N , whenever the frequency vector ν = (ν 1 , . . . , ν N ) ∈ R N is nonresonant, i.e. rationally independent. It is easy to verify that the dual group of T N is given by (T N ) * = {(θ 1 , . . . ,θ N ) → e 2πi(k 1 θ 1 +...+k N θ N ) : k ∈ Z N }.
Therefore, one possible choice for the cross section would be so ϕ(θ 1 , . . . ,θ N ) = e 2πiθ 1 . In this case, consecutive intersections of the flow and Σ would be separated by an interval of the length 1/ν 1 .

Almost periodic functions
The notion of almost periodic functions was introduced by H. Bohr as a generalization of strictly periodic functions [Boh25]. A function u ∈ C(R) is called (Bohr) almost periodic, if for any ǫ > 0 there is a relatively dense set of ǫ-almost-periods of this function. By this we mean, that for any ǫ > 0 there exists L = L(ǫ) such that any interval of length L contains at least on number T such that Later, Bochner [Boc27] gave an alternative but equivalent definition of this property: For a continuous function u, denote by u τ (t) the translated function u(t + τ). Then u is (Bohr) almost periodic if and only if every sequence u τ n n∈N of translations of u has a subsequence that converges uniformly. There are several other characterizations of almost periodicity, as well as generalizations due to Stepanov [Ste26], Weyl [Wey27] and Besicovitch [Bes26]. In this work we will only consider the notion depicted above and therefore call the corresponding functions just almost periodic (a.p.). We will however introduce one more way to describe a.p. functions using the framework of the previous section: Consider (Ω, ψ) as above and a function U ∈ C(Ω). Then, the function defined by is almost periodic. This can be verified easily with the alternative definition due to Bochner. Since U ∈ C(Ω), any sequence u τ n n∈N will be uniformly bounded and equicontinuous. Hence the Arzelà-Ascoli theorem guarantees the existence of a uniformly convergent subsequence. We will call any function obtainable in this manner representable over (Ω, ψ). Since the image of ψ is assumed to be dense, it is clear that the function U ∈ C(Ω) is uniquely determined by this relation. As an example take Ω S 1 , then ψ is periodic. Thus (2.1) gives rise to periodic functions. Conversely it is true, that any almost periodic function can be constructed this way. For this purpose we introduce the notion of hull. The hull H u of a function u is defined by where the closure is taken with respect to uniform convergence on the whole real line. Therefore if u is a.p., then H u is a compact metric space. If one uses the continuous extension of the rule u τ * u s = u τ+s ∀τ, s ∈ R onto all of H u as the group operation, then the hull becomes a commutative topological group with neutral element u.
These limits exist by Lemma 6.1 from the appendix. The continuity of both operations can be shown by a similar argument.) If we further define the flow ψ u (τ) = u τ , then the pair (H u , ψ u ) matches perfectly the setup of the previous section. Now, the representation formula (2.1) holds for U ∈ C(H u ) defined by This function is sometimes called the 'extension by continuity' of the almost periodic function u(t) to its hull H u . This construction is standard in the theory of a.p. functions and we refer the reader to [NS60] for a more detailed discussion.
For a function U : Ω → R let us introduce the derivative along the flow by Let C 1 ψ (Ω) be the space of continuous functions U : Ω → R such that ∂ ψ U exists for all ω ∈ Ω and ∂ ψ U ∈ C(Ω). The spaces C k ψ (Ω) for k ≥ 2 are defined accordingly. Let us also introduce the norm Lemma 2.2. Let U ∈ C(Ω) and u ∈ C(R) be such that u(t) = U(ψ(t)). Then we have u ∈ C 1 (R) and u ′ (t) is a.p. if and only if U ∈ C 1 ψ (Ω). One part of the equivalence is trivial. The proof of the other part can be found in [OT06,Lemma 13]. We also note that the derivative u ′ (t) of an almost periodic function is itself a.p. if and only if it is uniformly continuous. This, and many other interesting properties of a.p. functions are demonstrated in [Bes26].
Such functions are called quasi-periodic. In this case, ∂ ψ is just the derivative in the direction of ν ∈ R N . So if U is in the space C 1 (T N ) of functions in C 1 (R N ), which are 1-periodic in each argument, then Note however, that in general C 1 ψ (T N ) is a proper subspace of C 1 (T N ).

Haar measure and decomposition along the flow
It is a well known fact, that for every compact commutative topological group Ω there is a unique Borel probability measure µ Ω , which is invariant under the group operation, i.e.
µ Ω (D + ω) = µ Ω (D) holds for every Borel set D ⊂ Ω and every ω ∈ Ω. This measure is called the Haar measure of Ω. (This follows from the existence of the invariant Haar integral of Ω and the Riesz representation theorem. Proofs can be found in [Pon66] and [HR79], respectively.) For Example if Ω = S 1 we have where λ is the Lebesgue measure on R. Let ψ, Σ and Φ be as in section 2.1. Then Φ defines a decomposition Ω Σ × [0, S ) along the flow. Since Σ is a subgroup, it has a Haar measure µ Σ itself. Also the interval [0, S ) naturally inherits the probability measure As shown in [CT13], the restricted flow Φ : Σ × [0, S ) → Ω, Φ(σ, t) = σ · t also allows for a decomposition of the Haar measure µ Ω along the flow.
Lemma 2.4. The map Φ is an isomorphism of measure spaces, i.e.
holds for every Borel set B ⊂ Ω.
Before we prove this lemma, let us begin with some preliminaries. Consider the function χ : Since Φ is just the restricted flow, we have χ = id on Σ × [0, S ). This yields for every (σ, t) ∈ Σ × R, where ⌊·⌋ indicates the floor function. This representation shows that χ is measure-preserving on every strip Σ × [t, t + S ) of width S , since µ Σ and λ are invariant under translations in Σ and R, respectively. Moreover, the equality follows directly from the definition of χ.
Since µ Φ (Ω) = 1, this is a Borel probability measure. We will show that µ Φ is also invariant under addition in the group. For this purpose, let B ⊂ Ω be a Borel set and let ω 0 ∈ Ω. Then, by (2.4) we have (2.8) Therefore, µ Φ is a Borel probability measure on Ω which is invariant under group action.
3 A theorem about escaping sets 3.1 Measure-preserving embeddings From now on we will consider functions where D is an open set. We will call such a function measure-preserving embedding, if f is continuous, injective and furthermore holds for all Borel sets B ⊂ D, where λ denotes the Lebesgue measure of R. It is easy to show that under these conditions, f : D →D is a homeomorphism, whereD = f (D).
Since we want to use the iterations of f , we have to carefully construct a suitable domain on which these forward iterations are well-defined. We initialize This way f n is well-defined on D n . Clearly, f n is a measure-preserving embedding as well. Also inductively it can be shown that . . , f n (ω, r) ∈ D} and therefore D n+1 ⊂ D n ⊂ D for all n ∈ N. Initial conditions in the set correspond to complete forward orbits, i.e. if (ω 0 , r 0 ) ∈ D ∞ , then is defined for all n ∈ N. It could however happen that D ∞ = ∅ or even D n = ∅ for some n ≥ 2. The set of initial data leading to unbounded orbits is denoted by Complete orbits such that lim n→∞ r n = ∞ will be called escaping orbits. The corresponding set of initial data is

Almost periodic successor maps
Now, consider a measure-preserving embedding f : D ⊂ Ω × (0, ∞) → Ω × (0, ∞), which has the special structure f (ω, r) = (ω + ψ (F(ω, r)), r + G(ω, r)), where F, G : D → R are continuous. For ω ∈ Ω we introduce the notation ψ ω (t) = ω + ψ(t) = ω · t and define On this open set, consider the map f ω : Then f ω is continuous and meets the identity i.e. the following diagram is commutative: Therefore f ω is injective as well. Again we define D ω,1 = D ω and D ω,n+1 = f −1 ω (D ω,n ) to construct the set where the forward iterates (t n , r n ) = f n ω (t 0 , t 0 ) are defined for all n ∈ N. Analogously, unbounded orbits are generated by initial conditions in the set  These sets can also be obtained through the relations

Proof of Theorem 3.1
The proof of Theorem 3.1 is based on the fact, that almost all unbounded orbits of f are recurrent. In order to show this, we will apply the Poincaré recurrence theorem to the set U of unbounded orbits and the corresponding restricted map f U . We will use it in the following form [KO18, Lemma 4.2].
Lemma 4.1. Let (X, F , µ) be a measure space such that µ(X) < ∞. Suppose that there exists a measurable set Γ ⊂ X of measure zero and a map T : X \ Γ → X which is injective and so that the following holds: Since we can not guarantee that U has finite measure, we will also need the following refined version of the recurrence theorem due to Dolgopyat [Dol,Lemma 4.3].
Lemma 4.2. Let (X, F , µ) be a measure space and suppose that the map T : X → X is injective and such that the following holds: (c) there is a set A ∈ F such that µ(A) < ∞ with the property that almost all points from X visit A in the future.

Then for every measurable set B ⊂ X almost all points of B visit B infinitely many times in the future (i.e. T is infinitely recurrent).
For the sake of completeness let us state the proof.
Proof of Lemma 4.2. Let Γ ⊂ X be measurable such that µ(Γ) = 0 and all points of X \ Γ vist A in the future. Thus, the first return time r(x) = min{k ∈ N : T k (x) ∈ A} is welldefined for x ∈ X \ Γ. It induces a map S : X \ Γ → A defined by S (x) = T r(x) (x). The restriction S A\Γ is injective: Assume S (x) = S (y) for distinct points x, y ∈ A \ Γ and suppose r(x) > r(y), then T r(x)−r(y) (x) = y ∈ A is a contradiction to the minimality of r(x). It is also measure-preserving [EW11, cf. Lemma 2.43]. Now, consider a measurable set B ⊂ X and define B j = {y ∈ B \ Γ : r(y) ≤ j} as well as But since µ(A) < ∞ by assumption, the Poincaré recurrence theorem (Lemma 4.1) applies to A j . Thus we can find measurable sets Γ j ⊂ A j with measure zero, such that every point x ∈ A j \ Γ j returns to A j infinitely often (via S ). Now consider the set Then µ(F) = 0 and every point y ∈ B \ F returns to B infinitely often in the future. To see this, select j ∈ N such that r(y) ≤ j, i.e. y ∈ B j . Then x = S (y) ∈ A j \ Γ j . Hence there exist infinitely many k ∈ N so that k ≥ j and S k (x) ∈ A j . Let us fix one of these k. Then S k (x) = S (z) for some z ∈ B j . So in total we have Now, since k j=1 r(S j (y)) ≥ k + 1 > j ≥ r(z), this yields T m (y) = z ∈ B j ⊂ B, where m = k j=1 r(S j (y)) − r(z) ∈ N. One way to construct such a set A of finite measure is given by the next lemma [KO18]. It is based on the function W(ω, r) introduced in Theorem 3.1 and in fact is the only reason to assume the existence of W in the first place.
Then A has finite measure and every unbounded orbit of f enters A. More precisely, if (ω 0 , r 0 ) ∈ U, where U is from (3.1), and if (ω n , r n ) n∈N denotes the forward orbit under f , then there is K ∈ N so that (ω K , r K ) ∈ A.
Proof. First let us show that A has finite measure. By Fubini's theorem, holds for the sections A j,ω = {r ∈ (0, ∞) : (ω, r) ∈ A j }. Now, consider the diffeomorphism w ω : r → W(ω, r). Its inverse w −1 ω is Lipschitz continuous with constant β −1 , due to (3.4). But then, A j,ω = w −1 ω ((W j − ǫ j , W j + ǫ j )) implies λ(A j,ω ) ≥ 2β −1 ǫ j . Thus in total we have Next we will prove the recurrence property. To this end, let (ω 0 , r 0 ) ∈ U be fixed and denote by (ω n , r n ) the forward orbit under f . We will start with some preliminaries. Using (3.4) and the mean value theorem, we can findr such that Furthermore, by assumption we can find an index j 0 ≥ 2 such that W j 0 > max{W(ω 1 , r 1 ), k ∞ + max ω∈Ω W(ω,r), 2 k ∞ } and k 1 4γ Moreover we have lim sup n→∞ W(ω n , r n ) = ∞: Due to lim sup n→∞ r n = ∞, (3.4) implies W(ω n , r n ) ≥ β(r n − r 1 ) + W(ω n , r 1 ) for n sufficiently large. But then lim sup n→∞ W(ω n , r n ) = ∞ follows from the compactness of Ω. Now, since W(ω 1 , r 1 ) < W j 0 we can select the first index K ≥ 2 such that W(ω K , r K ) > W j 0 . So in particular this means W(ω K−1 , r K−1 ) ≤ W j 0 . Since (3.5) yields W(ω K , r K ) ≤ W(ω K−1 , r K−1 ) + k(r K−1 ), we can derive the following inequality: Then, the monotonicity of w ω K−1 implies r K−1 >r. Hence we can combine (4.2) with the previous estimate to obtain Now, we are ready to prove the theorem. We will assume that U ∅, since otherwise the assertion would be a direct consequence.
Step 1: Almost all unbounded orbits are recurrent. We will prove the existence of a set Z ⊂ U of measure zero such that if (ω 0 , r 0 ) ∈ U \ Z, then lim inf n→∞ r n < ∞.
In particular, we would have E ⊂ Z. To show this, we consider the restriction T = f U : U → U. This map is well-defined, injective and, like f , measure-preserving. We will distinguish three cases: In the first case Z = U is a valid choice. In case (ii) we can apply the Poincaré recurrence theorem (Lemma 4.1), whereas in case (iii) the modified version of Dolgopyat (Lemma 4.2) is applicable due to Lemma 4.3. Now, let us cover Ω × R by the sets B j = Ω × ( j − 1, j + 1) for j ∈ N. Then, for B j = B j ∩ U one can use the recurrence property to find sets Z j ⊂ B j of measure zero such that every orbit (ω n , r n ) n∈N starting in B j \ Z j returns to B j infinitely often. But this implies lim inf n→∞ r n ≤ r 0 + 2 < ∞. Therefore, the set Z = j∈N Z j ⊂ U has all the desired properties.
Step 2: The assertion is valid on the subgroup Σ ⊂ Ω. Since E ⊂ Z by construction, the inclusion holds for all ω ∈ Ω. To j ∈ Z we can consider the restricted flow It is easy to verify that just like Φ = Φ 0 of Lemma 2.4 those functions are isomorphisms of measure spaces. In other words, Φ j is bijective up to a set of measure zero, both Φ j and Φ −1 j are measurable, and for every Borel set B ⊂ Ω we have This clearly implies . Since Z has measure zero, (4.4) yields (µ Σ ⊗ λ 2 )(C j ) = 0. Next we consider the cross sections Then, λ 2 (C j,σ ) = 0 for µ Σ -almost all σ ∈ Σ follows from Fubini's theorem. So for every j ∈ Z there is a set M j ⊂ Σ with µ Σ (M j ) = 0 such that λ 2 (C j,σ ) = 0 for all σ ∈ Σ \ M j . Thus M = j∈Z M j has measure zero as well and and recalling that E σ ⊂ (ψ σ ×id) −1 (Z), we therefore conclude λ 2 (E σ ) = 0 for all σ ∈ Σ\M.

Statement and proof of the main result
We start with a rigorous description of the ping-pong map. To this end, let p be a forcing such that which sends a time t 0 of impact to the left plate x = 0 and the corresponding velocity v 0 > 0 immediately after the impact to their successors t 1 and v 1 describing the subsequent impact to x = 0. If we further denote byt ∈ (t 0 , t 1 ) the time of the particle's impact to the moving plate, then we can determinet =t(t 0 , v 0 ) implicitly through the equation since this relation describes the distance that the particle has to travel before hitting the moving plate. With that we derive a formula for the successor map: To ensure that this map is well defined, we will assume that This condition guarantees that v 1 is positive and also implies that there is a unique solutioñ t =t(t 0 , v 0 ) ∈ C 1 (R × (v * , ∞)) to (5.2). Thus we can take R × (v * , ∞) as the domain of the ping-pong map (5.3). Now, we are finally ready to state the main theorem.
We will give some further preliminaries before starting the actual proof. First we note, that the ping-pong map (t 0 , v 0 ) → (t 1 , v 1 ) is not symplectic. To remedy this defect, we reformulate the model in terms of time t and energy E = 1 2 v 2 . In these new coordinates the ping-pong map becomes P : (t 0 , E 0 ) → (t 1 , E 1 ), (5.7) . Since it has a generating function [KO10, Lemma 3.7], it is measure-preserving. Furthermore, from the inverse function theorem we can derive that P is locally injective. Note however, that in general P fails to be injective globally (see Appendix 6.2). Now, we will demonstrate that W(t 0 , E 0 ) = p(t 0 ) 2 E 0 acts as an adiabatic invariant for the ping-pong map. For this purpose we will cite the following lemma [KO10, Lemma 5.1]: Lemma 5.3. There is a constant C > 0, depending only upon p C 2 and a, b > 0 from (5.1), such that where (t 1 , E 1 ) = P(t 0 , E 0 ) denotes the ping-pong map for the forcing p, and So far we have depicted the case of a general forcing function p. Now we will replace p(t) by p ω (t) from (5.6) and study the resulting ping-pong map. First we note that due to P ∈ C 2 ψ (Ω) we have p ω ∈ C 2 (R). Also 0 < a ≤ p ω (t) ≤ b holds for all ω ∈ Ω by assumption. Furthermore, since ω · R lies dense in Ω it is In particular this means p ω C 2 (R) = P C 2 ψ (Ω) for all ω ∈ Ω. Therefore all considerations above apply with uniform constants. As depicted in Remark 5.2, also the threshold v * = 2 max{max ̟∈Ω ∂ ψ P(̟), 0} is uniform in ω. Finally, sincep ω (t) = ∂ 2 ψ P(ω + ψ(t)), the function ∆(t 0 , E 0 ) can be uniformly bounded by Hence, from Lemma 5.3 we obtain Lemma 5.4. There is a constant C > 0, uniform in ω ∈ Ω, such that where (t 0 , E 0 ) → (t 1 , E 1 ) denotes the ping-pong map P for the forcing function p ω (t).
Next we want to show that f is also measure-preserving. To this end, consider the maps g : (2.3). Then, the identity holds on D. This can be illustrated as follows: Recalling Lemma 2.4 and the fact that f ω has a generating function, it suffices to show that χ × id preserves the measure of any Borel set B ⊂ g (Φ −1 × id)(D) . Therefore, consider the sets Then we have as depicted in Section 2.3. Moreover, the injectivity of f implies the injectivity of χ × id on B and thus the sets (χ × id)(B k ) are mutually disjoint.
Remark 5.5. Let us also point out that the framework developed in the present paper can be applied to a lot of other dynamical systems. A famous example of such a system is given by the so called Littlewood boundedness problem. There, the question is whether solutions of an equationẍ + G ′ (x) = p(t) stay bounded in the (x,ẋ)-phase space if the potential G satisfies some superlinearity condition. In [Sch19] it is shown that the associated escaping set E typically has Lebesgue measure zero for G ′ (x) = |x| α−1 x with α ≥ 3 and a quasi-periodic forcing function p(t). Indeed, this result can be improved to the almost periodic case in a way analogous to the one presented here (for the ping-pong problem). where t ∈ R is arbitrary. Together this yields |u(τ n − s n + t) − u(τ m − s m + t)| < ǫ.
for all n, m ≥ N and t ∈ R, and thus proves the assertion.