Averaging almost periodic functions along exponential sequences
aa r X i v : . [ m a t h . N T ] A p r AVERAGING ALMOST PERIODIC FUNCTIONSALONG EXPONENTIAL SEQUENCES
MICHAEL BAAKE, ALAN HAYNES, AND DANIEL LENZ
Abstract.
The goal of this expository article is a fairly self-contained account of someaveraging processes of functions along sequences of the form ( α n x ) n ∈ N , where α is a fixedreal number with | α | > x ∈ R is arbitrary. Such sequences appear in a multitudeof situations including the spectral theory of inflation systems in aperiodic order. Due tothe connection with uniform distribution theory, the results will mostly be metric in nature,which means that they hold for Lebesgue-almost every x ∈ R . Introduction
A frequently encountered problem in mathematics and its applications is the study ofaverages of the form N P Nn =1 f ( x n ), where f is a function with values in C or, more generally,in some Banach space, and ( x n ) n ∈ N is a sequence of numbers in the domain of f . Quiteoften, an exact treatment of these averages is out of hand, and one resorts to the analysis ofasymptotic properties for large N . This, for instance, is common in analytic number theory;compare [19, 20, 1] and references therein. Equally important is the case where one canestablish the existence of a limit as N → ∞ , and then calculate it. This occupies a gooddeal of ergodic theory, where Birkhoff’s theorem and Kingman’s subadditive theorem providepowerful tools to tackle the problem; see [15, 38] for background.However, not all tractable cases present themselves in a way that is immediately accessibleto tools from ergodic theory. Also, depending on the nature of the underlying problem, onemight prefer a more elementary method, as Birkhoff-type theorems already represent a fairlyadvanced kind of ‘weaponry’. An interesting (and certainly not completely independent)approach is provided by the theory of uniform distribution of sequences, which essentiallygoes back to Weyl [39] and has emerged as a major tool for the study of function averages,in particular for functions that are periodic or defined on a compact domain; see [25, 17, 26]and references therein for more.In this contribution, we recall some of these concepts, with an eye on both methods (uni-form distribution and ergodic theory), and use the tools to treat averages of almost periodicfunctions along sequences where this makes sense, in particular along sequences of the form( α n x ) n ∈ N with ‘generic’ x ∈ R and a fixed number α ∈ R with | α | >
1. The first subtletythat we shall encounter here emerges when α is not an integer, which requires some care forfunctions that fail to be locally Riemann-integrable. The second subtlety occurs when weextend our considerations to almost periodic functions. MICHAEL BAAKE, ALAN HAYNES, AND DANIEL LENZ
While the latter extension represents a relatively simple step beyond periodic functions aslong as one retains almost periodicity in the sense of Bohr, matters become more involvedwhen singularities occur or weaker notions of almost periodicity are needed. Below, we shalldiscuss some extensions of this kind that are relevant in practice; compare [18] for somerelated results. Let us note that some of the notions and concepts used below are studied inmuch greater generality in [29, 36].Before we begin our exposition, let us mention that averages of 1-periodic functions are oftenjust the first step in the study of Riesz–Raikov sums, that is, sums of the form P n − k =0 f ( α k t ).Kac’s investigation for α = 2 in [22] and Takahashi’s refined and generalised analysis [37] areearly examples that consider limits (in a law of large numbers scaling) as well as distributions(in a central limit theorem scaling, when R f ( t ) d t = 0). This led to a more elaborate deriva-tion of central limit theorems for Riesz–Raikov sums along exponential sequences; compare[31, 27, 33] and references therein.Below, we are mainly interested in the Birkhoff-type averages, with a focus on functionsthat fail to be periodic, but still have some repetitivity structure in the form of a suitablealmost periodicity. In this sense, we have selected one particular aspect of Riesz–Raikov sumsthat appears in the theory of aperiodic order [3, 10, 2].2. Preliminaries and general setting
As far as possible, we follow the general (and fairly standard) notation from [3, Ch. 1],wherefore only deviations or extensions will be mentioned explicitly. In particular, we willuse the Landau symbols O and O for the standard asymptotic behaviour of real- or complex-valued functions; compare [1, 19] for definitions and examples.When two sets A, B ⊆ R are given, we denote their Minkowski sum as A + B := { a + b : a ∈ A, b ∈ B } . In particular, if the point set S ⊂ R is locally finite and ε >
0, we use S + ( − ε, ε ) for theopen subset of R that emerges from S as S x ∈ S ( x − ε, x + ε ). Note that its complement in R is then a closed set (possibly empty).Below, we frequently talk about results of metric nature, where Lebesgue measure λ on R is our reference measure. When a statement is true for almost every x ∈ R with respect toLebesgue measure, we will simply say that it holds for a.e. x ∈ R . Likewise, when we speakof a null set, we mean a null set with respect to Lebesgue measure.Recall that a sequence ( x n ) n ∈ N of real numbers is called uniformly distributed modulo a, b with 0 a < b
1, we havelim N →∞ N card (cid:16) [ a, b ) ∩ (cid:8) h x i , . . . , h x N i (cid:9)(cid:17) = b − a, VERAGING ALMOST PERIODIC FUNCTIONS 3 where h x i denotes the fractional part of x ∈ R . We refer to [25, 11] for general background.Recall that a function f on R is 1- periodic if f ( x + 1) = f ( x ) holds for all x ∈ R . Onefundamental result, due to Weyl [39], can now be formulated as follows; see also [20, Thm. 5.3]. Lemma 2.1 ( Weyl’s criterion ) . For a sequence ( x n ) n ∈ N of real numbers, the following prop-erties are equivalent. (1) The sequence is uniformly distributed modulo . (2) For every complex-valued, -periodic continuous function f , one has lim N →∞ N N X n =1 f ( x n ) = Z f ( x ) d x. (3) The relation from (2) holds for every -periodic function that is locally Riemann-integrable. (4) The relation lim N →∞ N N X n =1 e π i kx n = δ k, holds for every k ∈ Z . (cid:3) Let us note in passing that the equivalence of conditions (1) and (2) can also be understoodin terms of systems of almost invariant integrals and Eberlein’s ergodic theorem. Thesenotions are reviewed and studied in some detail in [36].
Remark 2.2.
Weyl’s criterion (which is also known as Weyl’s lemma) is an important toolfor calculating the average of a locally Riemann-integrable periodic function along a uniformlydistributed sequence. In fact, a 1-periodic function is locally Riemann-integrable if and onlyif the Birkhoff average converges for every sequence that is uniformly distributed modulo 1;compare [16] as well as [20, p. 123].Conversely, the integral of a Riemann-integrable function can be approximated by averagesalong uniformly distributed sequences. This is a standard method in numerical integration,in particular for higher-dimensional integrals; see [21, 26] and references therein for more. ♦ There is an abundance of known results on uniformly distributed sequences and their finerproperties; we refer to [25] for the classic theory and to [11] and references therein for morerecent developments. Here, we are particularly interested in one specific class of sequences,for which the uniform distribution is well known; compare [11, Thms. 1.7 and 1.10] as well as[15, Sec. 7.3, Thm. 1] or [25, Cor. 1.4.3 and Exs. 1.4.3].
Fact 2.3.
Consider the sequence ( α n x ) n ∈ N . For fixed α ∈ R with | α | > , it is uniformlydistributed modulo for a.e. x ∈ R . For fixed = x ∈ R , the sequence is uniformly distributedmodulo for a.e. α ∈ R with | α | > . (cid:3) Since we use { x } for singleton sets, we resort to the less common notation h x i for the fractional part of x in order to avoid misunderstandings. MICHAEL BAAKE, ALAN HAYNES, AND DANIEL LENZ
Below, we will mainly be concerned with the first case, where a fixed α with α > | α | > α = q > q n x ) n ∈ N is uniformlydistributed modulo 1 if and only if x is a normal number [11] in base q , which means thatthe q -ary expansion of x contains all possible finite substrings in the digit set { , , . . . , q − } in such a way that any substring of length ℓ has frequency 1 /q ℓ . In the Lebesgue sense, a.e. x ∈ R is normal with respect to all integer bases [11, Thm. 4.8], but it is a hard problem todecide on normality for any given number. Remark 2.4.
Consider a sequence ( u n ) n ∈ N of real numbers such that inf n = m | u n − u m | > u n x ) n ∈ N is uniformly distributed modulo 1 for a.e. x ∈ R . In fact, it is a rather direct consequence that, for any k ∈ N , ℓ ∈ N and any realnumber L >
0, the arithmetic progression sequence ( u km + ℓ x ) m ∈ N is uniformly distributedmodulo L for a.e. x ∈ R . This is the total Bohr ergodicity of the sequence ( u n ) n ∈ N asintroduced in [18, Def. 2.1]. Clearly, u n = α n with | α | > ♦ As soon as we leave the realm of periodic functions that are locally Riemann-integrable, thedesired averaging statements will need some finer properties of our sequences ( α n x ) n ∈ N , wherewe assume | α | > discrepancy of a sequence ( x n ) n ∈ N is quantified in terms of the first N elements of the sequence (taken modulo 1), namely by the number D N := sup a
Let α ∈ R with | α | > be given. Then, for any fixed ε > , the discrepancy of thesequence ( α n x ) n ∈ N , for a.e. x ∈ R , asymptotically is D N = O (cid:0) log( N ) (cid:1) + ε √ N as N → ∞ . (cid:3) VERAGING ALMOST PERIODIC FUNCTIONS 5
Next, we need a
Diophantine approximation property. If ∅ = Y ⊂ R is a uniformly discretepoint set, compare [3, Sec. 2.1], we can definedist( x, Y ) := min y ∈ Y | x − y | as the distance of x ∈ R from Y . Now, one can state the following metric ‘non-approximation’result, which is a versatile generalisation of the classic situation with Y = Z . Lemma 2.6.
Let α ∈ R with | α | > be given, and let Y ⊂ R be a non-empty, uniformlydiscrete point set. Further, fix some ε > . Then, for a.e. x ∈ R , the inequality dist( α n − x, Y ) > n ε holds for almost all n ∈ N , by which we mean that it holds for all natural numbers except atmost finitely many.Proof. The statement is trivial when Y is a finite point set, so let us assume that Y isunbounded. In this case, one still has δ := inf (cid:8) | x − y | : x, y ∈ Y , x = y (cid:9) > , due to the assumed uniform discreteness of Y . Consequently, the number of points of Y inan arbitrary interval [ a, b ] with a b satisfies(2.1) card (cid:0) Y ∩ [ a, b ] (cid:1) h b − aδ i , where [ . ] is the Gauß bracket.Let m ∈ Z be arbitrary, but fixed, and consider I m = [ m, m + 1]. With R = S m ∈ Z I m ,it suffices to show that our claim fails at most for a null set within the interval I m , as thecountable union of null sets is still a null set.Choose ε > n ∈ N , consider the set A ( m ) n := n x ∈ I m : dist( α n − x, Y ) < n ε o . It is clearly measurable, and its measure, since | α | >
1, can be estimated as λ (cid:0) A ( m ) n (cid:1) = 1 | α | n − λ n z ∈ α n − I m : dist( z, Y ) < n ε o | α | n − n ε (cid:18) h | α | n − δ i(cid:19) = O (cid:16) n ε (cid:17) , where the second step is a consequence of Eq. (2.1). We thus know that there is a C > λ (cid:0) A ( m ) n (cid:1) C/n ε for all n ∈ N .Now, we have 0 X n > λ (cid:0) A ( m ) n (cid:1) C X n > n ε , MICHAEL BAAKE, ALAN HAYNES, AND DANIEL LENZ where the second sum is convergent, and thus also the first. Then, Cantelli’s lemma tells usthat E ( m ) ∞ := (cid:8) x ∈ I m : x ∈ A ( m ) n for infinitely many n ∈ N (cid:9) is indeed a null set, which is what we needed to show. (cid:3) Remark 2.7.
Though immaterial for the proof, it is often useful in an application to alsoremove all x ∈ R with Y ∩ { α n − x : n ∈ N } 6 = ∅ , which constitutes a null set because it isclearly countable or even finite. ♦ The lower bound in Lemma 2.6 can be replaced by the values of a more general, non-negative arithmetic function, ψ ( n ) say, provided one also has the summability condition P n ∈ N ψ ( n ) < ∞ . When this sum diverges, the situation changes. Indeed, for instance if α = 2 and Y = Z , there is then a set X ⊂ R of full measure such that, for x ∈ X , thedistance of 2 n − x from the nearest integer is smaller than n for infinitely many n ∈ N ; see [4]for a more general result in this direction. Moreover, one cannot do better than using some ε > Averaging periodic functions
Let us first state a result that emerges from an application of Weyl’s criterion to the specialtype of sequences we are interested in.
Fact 3.1.
Let f : R −−→ C be a continuous or, more generally, a locally Riemann-integrablefunction that is L -periodic, so f ( x + L ) = f ( x ) holds for some fixed L > and all x ∈ R . If α is a real number with | α | > , one has lim N →∞ N N − X n =0 f ( α n x ) = 1 L Z L f ( y ) d y for a.e. x ∈ R .Proof. Since any L -periodic continuous function is also locally Riemann-integrable, it sufficesto consider the latter class. Define a new function g by g ( x ) := f ( Lx ), which clearly is1-periodic and locally Riemann-integrable. Now, we have1 N N − X n =0 f ( α n x ) = 1 N N − X n =0 g (cid:0) α n xL (cid:1) , where (cid:0) α n xL (cid:1) n ∈ N is uniformly distributed modulo 1 for a.e. xL ∈ R , and hence also for a.e. x ∈ R , by Fact 2.3. Consequently, Weyl’s criterion from Lemma 2.1 tells us that1 N N − X n =0 f ( α n x ) N →∞ −−−−→ Z g ( z ) d z = 1 L Z L f ( y ) d y holds for all such cases, which means for a.e. x ∈ R as claimed. (cid:3) This being the ‘easy half’ of the Borel–Cantelli lemma, which goes back to Cantelli, we follow [11, App. C]in our terminology, and also refer to this reference for a proof.
VERAGING ALMOST PERIODIC FUNCTIONS 7
Note that one can rewrite Fact 3.1 with the mean of f , because M ( f ) := lim T →∞ T Z T − T f ( y ) d y = lim T →∞ T Z a + Ta f ( y ) d y = 1 L Z L f ( y ) d y holds for every L -periodic function that is locally Riemann-integrable, where the limit clearlyis uniform in a ∈ R . Example 3.2.
Fix k ∈ R and consider the trigonometric monomial defined by ψ k ( x ) = e π i kx .Unless k = 0, in which case ψ ≡
1, the function ψ k has period | k | >
0. For α ∈ R with | α | >
1, Fact 3.1 implies that1 N N − X n =0 ψ k ( α n x ) N →∞ −−−−→ M ( ψ k ) = ( , k = 0 , , otherwise , holds for a.e. x ∈ R .More generally, if ( u n ) n ∈ N with inf n = m | u n − u m | > α n x replaced by u n x . ♦ It is clear from the proof of Fact 3.1 that, for periodic functions, it suffices to consider thecase L = 1 without loss of generality, as we do from now on. Our next step shows that, for α ∈ Z , one can go beyond the class of 1-periodic functions that are locally Riemann-integrable. Lemma 3.3.
Consider a function f ∈ L ( R ) that is -periodic. Fix q ∈ Z with | q | > .Then, for a.e. x ∈ R , one has N N − X n =0 f ( q n x ) N →∞ −−−−→ Z f ( y ) d y = M ( f ) . Proof.
Since q ∈ Z , we may view the average as a Birkhoff sum for the dynamical systemon [0 ,
1] defined by the mapping x qx mod 1. It is well known that Lebesgue measureis invariant and ergodic for this system, compare [12] and references therein, wherefore wemay employ Birkhoff’s ergodic theorem [38] to f , which is Lebesgue-integrable on [0 ,
1] byassumption, and our claim follows. (cid:3)
Note that the exceptional set, for which the limit differs or does not exist, may depend on f when the latter fails to be continuous. In fact, there clearly is no uniformly distributedsequence that will work for all f ∈ L ( R ). Still, the result of Lemma 3.3 suggeststhat something more general than Fact 3.1 might also be true when our multiplier α fails tobe an integer. However, we cannot apply the ‘trick’ with Birkhoff’s ergodic theorem when α Z . This is due to the fact that the sequence ( h α n x i ) n ∈ N , which is uniformly distributedon [0 ,
1) for a.e. x ∈ R by Fact 2.3, does no longer agree with the orbit of x under themapping T defined by x αx mod 1. The latter, for a.e. x ∈ R , follows the distribution ofthe (ergodic) R´enyi–Parry measure [32, 30] for α , which is of the form h α λ with h α beingLebesgue-integrable on [0 , α Z , the measures λ and h α λ are still equivalent asmeasures, but different; see [12] and references therein for more. MICHAEL BAAKE, ALAN HAYNES, AND DANIEL LENZ
Example 3.4.
To illustrate the difference, consider α = τ = (cid:0) √ (cid:1) , which is one of thesimplest examples in this context. When f is 1-periodic and locally Riemann-integrable, weget 1 N N − X n =0 f ( τ n x ) N →∞ −−−−→ Z f ( x ) d x = M ( f )for a.e. x ∈ R by Weyl’s criterion (Lemma 2.1).In comparison, let T be defined by x τ x mod 1 on [0 , x ∈ [0 , T n x ) n ∈ N follow the distribution given by the piecewise constant function [32, Ex. 4] h τ ( x ) = √ , ≤ x < τ , √ , τ ≤ x < . Since T is ergodic for the measure h τ λ , Birkhoff’s theorem tells us that, for any Lebesgue-integrable function f on [0 , N →∞ N N − X n =0 f ( T n x ) = Z f ( x ) h τ ( x ) d x for a.e. x ∈ [0 , M ( f ).Moreover, since the sequences ( h τ n x i ) n ≥ and ( T n x ) n ≥ are not easily relatable, one cannotinfer the convergence of averages along the exponential sequence from those along the orbitsunder T . ♦ Let us now extend Fact 3.1 beyond Riemann-integrable functions by stating one version ofSobol’s theorem [34, Thm. 1].
Theorem 3.5.
Let α ∈ R with | α | > be fixed, and consider a -periodic function f ∈ L ( R ) that fails to be locally Riemann-integrable. Assume that there is a finite set F ⊂ [0 , such that f , for every δ > , is Riemann-integrable on the complement of F + ( − δ, δ ) in [0 , . Assume further that, for every z ∈ F , there is a δ z > such that f is differentiable onthe punctured interval ( z − δ z , z + δ z ) \ { z } and that, for any s > , V N ( z, s ) := Z z − Ns z − δ z | f ′ ( x ) | d x + Z z + δ z z + Ns | f ′ ( x ) | d x = O (cid:0) N s − η (cid:1) holds for some η = η ( z ) > as N → ∞ .Then, for a.e. x ∈ R , one has N N − X n =0 f ( α n x ) N →∞ −−−−→ Z f ( y ) d y = M ( f ) . Sketch of proof.
Since F is finite, we may choose 0 < δ min z ∈ F δ z small enough such thatthe open sets ( z − δ, z + δ ) with z ∈ F are disjoint. By writing f as a sum of a locally Riemann-integrable function (such as the restriction f ( δ ) of f to the complement of Z + F + ( − δ, δ ))and r = card( F ) ‘problematic’ terms, the latter supported on ( z − δ, z + δ ) with z ∈ F , itis clear that our claim follows if we can deal with one of these problematic terms. So, select VERAGING ALMOST PERIODIC FUNCTIONS 9 one z ∈ F . Without loss of generality, we may assume that ( z − δ, z + δ ) ⊂ [0 , f and f ( δ ) are 1-periodic.One can now repeat the original proof from [34], or the more extensive version in [21,Sec. 2]. Here, the validity of the convergence claim emerges from the observation that, fora.e. x ∈ R , the number h α n − x i does not come closer to z than 1 /n ε , for any fixed ε > n ∈ N except at most finitely many. This follows from Lemma 2.6 with Y = z + Z .Now, V N ( z, ε ) = O (cid:16) N ε − η (cid:17) for some η > η does not depend on ε , we are still free to choose ε > ϑ := η − ε > z are properly ‘counterbalanced’ by the discrepancy of ( α n x ) n ∈ N , where we invoke Fact 2.5with the ε just chosen. One obtains(3.1) D N · V N ( z, ε ) = O (cid:18) (cid:0) log( N ) (cid:1) + ε N ϑ (cid:19) = O (1) , which is a sufficient criterion for the claimed convergence because1 N N − X n =0 f ( δ ) ( α n x ) N →∞ −−−−→ M (cid:0) f ( δ ) (cid:1) holds for a.e. x ∈ R , while the Birkhoff average of f − f ( δ ) is controlled by Eq. (3.1) and tendsto 0 as δ ց (cid:3) Remark 3.6.
The assumption that F in Theorem 3.5 is a finite set implies δ := min z ∈ F δ z > z ∈ F η ( z ) >
0. Later, we will replace this setting by a suitable compactnessassumption to extend the result of this theorem to almost periodic functions. ♦ Remark 3.7.
The differentiability assumption for f near the ‘bad’ points is convenient, butnot necessary. It can be replaced by the requirement that the total variation of f on sets ofthe form ( z − δ, z − N − s ] ∪ [ z + N − s , z + δ ) behaves as stated for V N ( z, s ); compare [34, 21]. ♦ As mentioned earlier, results of this type are also of interest for the numerical calculationof integrals, for instance with methods of (quasi-) Monte Carlo type. In our context, animportant question is how to extend Riesz–Raikov sums and Birkhoff averages to functionsthat fail to be periodic, but possess some repetitivity structure instead.4.
Averaging almost periodic functions
At this point, we need to recall some basic definitions and results from the theory of almostperiodic functions in the sense of Bohr [9], where we refer to [3, Sec. 8.2] for a short summary,to [24, Sec. VI.5] or [14] for comprehensive expositions, and to [28, Sec. 41] for a more generaland abstract setting (including non-Abelian groups).
Recall that f ∈ C ( R ) is called almost periodic in the sense of Bohr if, for any ε >
0, theset of ε -almost periods P ε := (cid:8) t ∈ R : k f − T t f k ∞ < ε (cid:9) is relatively dense in R . Here, (cid:0) T t f (cid:1) ( x ) := f ( x − t ) defines the t -translate of f . Any continuousperiodic function is almost periodic in this sense, as is any trigonometric polynomial. AnyBohr-almost periodic function is bounded and uniformly continuous. In fact, the k . k ∞ -closureof the (complex) algebra of trigonometric polynomials is precisely the space of all Bohr-almostperiodic functions [9].For comparison, f ∈ C ( R ) is called almost periodic in the sense of Bochner (for k . k ∞ , tobe precise) if the translation orbit { T t f : t ∈ R } is precompact in the k . k ∞ -topology. Thefundamental relation among these notions can be summarised as follows; see [3, Prop. 8.2] aswell as [24, 14]. Fact 4.1.
For f ∈ C ( R ) , the following properties are equivalent. (1) f is Bohr-almost periodic, i.e., P ε is relatively dense for any ε > ; (2) f is Bochner-almost periodic for k . k ∞ , i.e., the orbit { T t f : t ∈ R } is precompact inthe k . k ∞ -topology; (3) f is the limit of a sequence of trigonometric polynomials, with uniform convergenceof the sequence on R . (cid:3) In view of these relations, we follow [6] and speak of uniformly almost periodic functions from now on when we refer to this class. If misunderstandings are unlikely, we will drop theattribute ‘uniformly’. Let us elaborate a little on part (3) of Fact 4.1. If f is almost periodic,its mean(4.1) M ( f ) = lim T →∞ T Z a + Ta − T f ( x ) d x exists for any a ∈ R , is independent of a , and the convergence is uniform in a ; compare [36]for a more detailed discussion of this concept. When we need to emphasise the role of a for more general types of functions (say without uniformity of the limit in a ), we will write M ( f ; a ).The Fourier–Bohr coefficient of an almost periodic function f at k ∈ R is given by a ( k ) = M (cid:0) e − π i k ( . ) f (cid:1) . It exists for any k ∈ R , and differs from 0 for at most countably many values of k . Any k ∈ R with a ( k ) = 0 is called a frequency of f . If { k ℓ } is the set of frequencies of f , there isa sequence of trigonometric polynomials of the form(4.2) P ( m ) ( x ) = n m X ℓ =1 r ( m ) ℓ a ( k ℓ ) e π i k ℓ x that converge uniformly to f on R as m → ∞ . Here, the numbers r ( m ) ℓ , which are known asconvergence enforcing numbers, depend on m and k ℓ , but not on a ( k ℓ ), and can be chosen asrational numbers [14, Thm. I.1.24]. VERAGING ALMOST PERIODIC FUNCTIONS 11
To approach averages of almost periodic functions, it is thus more than natural to beginwith the averages of trigonometric polynomials. We formulate the next result for more generalsequences than the exponential ones from above.
Proposition 4.2.
Let P m be a ( complex ) trigonometric polynomial of the form P m ( x ) = a + m X ℓ =1 a ℓ e π i k ℓ x , with coefficients a ℓ ∈ C and distinct non-zero frequencies k , . . . , k m . Further, let ( u n ) n ∈ N be a sequence of real numbers such that inf n = m | u n − u m | > . Then, for a.e. x ∈ R , one has lim N →∞ N N − X n =0 P m ( u n x ) = M ( P m ) = a . In particular, this holds for u n = α n with α ∈ R and | α | > .Proof. The claim is obvious for m = 0, where the polynomial is constant. The case m = 1with a = 0, where P m is a monomial, is Example 3.2 from above. So, for a general P m , theclaim is true for each summand individually, with an exceptional set E ( k ℓ ) of measure 0 for ℓ ≥
1. Since S mℓ =1 E ( k ℓ ) is still a null set, the statement on the limit is clear, while its valuefollows from a simple calculation with the mean; compare Example 3.2. (cid:3) Before we proceed, let us recall the following useful property of the mean.
Lemma 4.3.
Let ( g n ) n ∈ N be a sequence of complex-valued, but not necessarily continuous,functions on R that converge uniformly to a function f . Assume further that the mean M ( g n ) exists for all n ∈ N . Then, also M ( f ) exists, and lim n →∞ M ( g n ) = M ( f ) . In particular,one has M ( f ) = lim T →∞ T Z a + Ta − T f ( x ) d x for any fixed a ∈ R . When the convergence of the means M ( g n ) = M ( g n ; a ) is uniform in a ,then so is the convergence of M ( f ) .Proof. The assumed uniform convergence also means that ( g n ) n ∈ N is a Cauchy sequence inthe k . k ∞ -topology. Fix ε > n = n ( ε ) such that k g n − f k ∞ < ε as well as k g n − g m k ∞ < ε holds for all n, m > n . Then, for any T >
0, one has(4.3) 12 T (cid:12)(cid:12)(cid:12)(cid:12)Z T − T (cid:0) g n ( x ) − g m ( x ) (cid:1) d x (cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13) g n − g m (cid:13)(cid:13) ∞ < ε for all n, m > n , which implies (cid:12)(cid:12) M ( g n ) − M ( g m ) (cid:12)(cid:12) (cid:13)(cid:13) g n − g m (cid:13)(cid:13) ∞ + (cid:12)(cid:12)(cid:12)(cid:12) M ( g n ) − T Z T − T g n ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) M ( g m ) − T Z T − T g m ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12) . Consequently, (cid:12)(cid:12) M ( g n ) − M ( g m ) (cid:12)(cid:12) < ε for all sufficiently large T due to our assumption onthe existence of the means M ( g n ). Note that, although T may depend on m and n , the above 3 ε -estimate still works as a consequence of Eq. (4.3). The sequence (cid:0) M ( g n ) (cid:1) n ∈ N is thusCauchy, hence convergent, with limit M , say.Now, choose n > n large enough such that also | M ( g n ) − M | < ε holds, fix an arbitrary a ∈ R , and consider (cid:12)(cid:12)(cid:12)(cid:12) T Z a + Ta − T f ( x ) d x − M (cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13) f − g n (cid:13)(cid:13) ∞ + (cid:12)(cid:12) M ( g n ) − M (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) T Z a + Ta − T g n ( x ) d x − M ( g n ) (cid:12)(cid:12)(cid:12)(cid:12) < ε, where the last step holds for all sufficiently large T by assumption. This derivation implieslim T →∞ T Z a + Ta − T f ( x ) d x = M = lim n →∞ M ( g n ) , which is independent of a ∈ R , and M ( f ) = M is the claimed mean of f .When, in addition, the means of the functions g n exist uniformly in a , our 3 ε -argumentalso implies that the convergence of M ( f ; a ) is uniform in a ∈ R as claimed. (cid:3) This enables us to formulate the following result.
Theorem 4.4.
Let α ∈ R with | α | > be given, and let f be a Bohr-almost periodic functionon R . Then, for a.e. x ∈ R , one has lim N →∞ N N − X n =0 f ( α n x ) = M ( f ) . Proof.
Let ( g n ) n ∈ N be a sequence of trigonometric polynomials that converge uniformly to f .As is well known, compare [14], and is a rather direct consequence of Eq. (4.2), the sequencecan be chosen such that the frequency sets { k j : 1 j m n } of the g n are nested. ByProposition 4.2, we know that, for every n ∈ N ,lim N →∞ N N − X ℓ =0 g n ( α ℓ x ) = M ( g n )holds for a.e. x ∈ R , where we denote the excluded null set by E n . By construction, we have E n ⊆ E n +1 , and E := S n ∈ N E n is still a null set.Define the Birkhoff average of ϕ at x as S N ( ϕ, x ) = N P N − n =0 ϕ ( α n x ), and fix some ε > n = n ( ε ) such that k f − g n k ∞ < ε for all n > n , which is possible under ourassumptions. Now, for any fixed x ∈ R \ E , we can estimate (cid:12)(cid:12) S N ( f, x ) − M ( f ) (cid:12)(cid:12) (cid:12)(cid:12) S N ( f − g n , x ) (cid:12)(cid:12) + (cid:12)(cid:12) S N ( g n , x ) − M ( g n ) (cid:12)(cid:12) + (cid:12)(cid:12) M ( g n ) − M ( f ) (cid:12)(cid:12) where, independently of N , (cid:12)(cid:12) S N ( f − g n , x ) (cid:12)(cid:12) S N (cid:0) | f − g n | , x (cid:1) k f − g n k ∞ < ε for any n > n . The third term on the right-hand side of the previous estimate is smallerthan ε for sufficiently large n as a consequence of Lemma 4.3, while the middle term, under VERAGING ALMOST PERIODIC FUNCTIONS 13 our assumptions, is bounded by ε for sufficiently large N , which we are still free to choose.This 3 ε -argument thus establishes the claim. (cid:3) Let us mention in passing that Theorem 4.4 still holds if α n , as before, is replaced by thenumbers u n of a sequence as described in Remark 2.4.At this point, to go any further, we need to extend the class of functions we consider. Thisis motivated by the fact that uniform almost periodicity is often too restrictive. In particular,in various examples from dynamical systems theory, one encounters averages over functionsthat fail to be bounded, and hence cannot be uniformly almost periodic. Being unbounded,such functions cannot be locally Riemann-integrable either, though they might still admitimproper Riemann integrals or be locally Lebesgue-integrable.It would be natural to investigate the question in the setting of weakly almost periodicfunctions, as introduced in [36], which seems possible as well. However, the above remarksindicate that one needs results also for functions that violate continuity. This suggests to usethe wider class of almost periodic functions in the sense of Stepanov [35], which relate tolocally Lebesgue-integrable functions like uniform almost periodic functions do to continousfunctions. The new norm on L ( R ) is given by k f k S = sup x ∈ R L Z x + Lx | f ( y ) | d y, where L > L , it is most convenient to choose L = 1, as we do from now on. Now, a locallyLebesgue-integrable function f is called almost periodic in the sense of Stepanov , or S-almostperiodic for short, if, for any ε >
0, the set P S ε of ε -almost periods of f for k . k S is relativelydense. The analogue of Fact 4.1 then reads as follows (we omit a proof because it works thesame way as in the previous case; compare [14]). Fact 4.5.
For f ∈ L ( R ) , the following properties are equivalent. (1) f is S -almost periodic, i.e., P S ε is relatively dense for any ε > ; (2) f is Bochner-almost periodic for k . k S , i.e., the orbit { T t f : t ∈ R } is precompact inthe k . k S -topology; (3) f is the k . k S -limit of a sequence of trigonometric polynomials. (cid:3) Let us note in passing that every locally integrable function f on R may be viewed asa translation bounded measure (where f is the Radon–Nikodym density relative to λ ). Indoing so, the Stepanov norm is induced by the k . k [0 , -norm for measures as discussed in [36].This implies that a function f ∈ L ( R ) is S-almost periodic if and only if the measure f λ isnorm-almost periodic in the sense of [5, 36].Every uniformly almost periodic function is S-almost periodic, which also means (via part(3) of Fact 4.5) that any S-almost periodic function can be k . k S -approximated by uniformly The widely used modern version of the name is V.V. Stepanov, while the author used W. Stepanoff in hisoriginal articles. almost periodic functions. In other words, the class of all S-almost periodic functions canequivalently be described as the k . k S -closure of the (complex) algebra of trigonometric poly-nomials or as that of the class of uniformly almost periodic functions. Moreover, the space ofS-almost periodic functions is complete in the k . k S -norm, and k f k S = 0 means f = 0 in theLebesgue sense, so f ( x ) = 0 for a.e. x ∈ R ; see [7] for details. Remark 4.6. If f is S-almost periodic, its mean exists. In fact, observe that, for all S-almostperiodic functions f, g and for any a ∈ R , one has12 T (cid:12)(cid:12)(cid:12)(cid:12)Z a + Ta − T (cid:0) f ( x ) − g ( x ) (cid:1) d x (cid:12)(cid:12)(cid:12)(cid:12) T ]2 T (cid:13)(cid:13) f − g (cid:13)(cid:13) S . Now, it is immediate that the statement of Lemma 4.3 still holds if uniform convergenceis replaced by k . k S -convergence. This then gives the desired existence of means because,by Fact 4.5(3), we can k . k S -approximate any S-almost periodic function with trigonometricpolynomials for which the mean clearly exists. ♦ As an aside, we mention the following interesting connection.
Lemma 4.7.
Let f be an S -almost periodic function, and let δ > be arbitrary, but fixed.Then, the function f δ defined by f δ ( x ) = 12 δ Z x + δx − δ f ( y ) d y is continuous and uniformly almost periodic. Moreover, lim δ ց f δ = f in the k . k S -topology.Proof. Assume δ (the argument for δ > is analogous), and let t be a (2 δε )-almostperiod of f for k . k S . Now, (cid:12)(cid:12) f δ ( t + x ) − f δ ( x ) (cid:12)(cid:12) = 12 δ (cid:12)(cid:12)(cid:12)(cid:12)Z x + δx − δ f ( t + y ) − f ( y ) d y (cid:12)(cid:12)(cid:12)(cid:12) δ Z x − δ +1 x − δ (cid:12)(cid:12) f ( t + y ) − f ( y ) (cid:12)(cid:12) d y k f − T t f k S δ < ε which implies that t is an ε -almost period of f δ for k . k ∞ . Via part (1) of Fact 4.5, we concludethat f δ satisfies part (1) of Fact 4.1, and thus is uniformly almost periodic. As such, f δ isalso uniformly continuous.For the second claim, we refer to the original proof in [7], which uses an approximationargument that is based on the effect that a ‘convolution mollifier’ has on a locally Lebesgue-integrable function. (cid:3) The main extension of Theorem 3.5 can be stated as follows.
Theorem 4.8.
Let α ∈ R with | α | > be fixed, and let f ∈ L ( R ) be an S -almost periodicfunction. Assume now that there is a uniformly discrete set Y ⊂ R such that f , for every δ > , is locally Riemann-integrable on the complement of Y + ( − δ, δ ) . Assume further that VERAGING ALMOST PERIODIC FUNCTIONS 15 there is a δ ′ > such that, for any z ∈ Y , f is differentiable on the punctured interval ( z − δ ′ , z + δ ′ ) \ { z } and that, for any s > and with V N ( z, s ) as defined in Theorem . , sup z ∈ Y V N ( z, s ) = O (cid:0) N s − η (cid:1) holds for some η > as N → ∞ .Then, for a.e. x ∈ R , one has lim N →∞ N N − X n =0 f ( α n x ) = M ( f ) , where the mean exists because f is S -almost periodic.Sketch of proof. Without loss of generality, we may assume that δ ′ is small enough so thatthe open intervals ( z − δ ′ , z + δ ′ ) with z ∈ Y are disjoint. Now, Lemma 2.6 guarantees thatthe sequence ( α n − x ) n ∈ N , for a.e. x ∈ R , does not come closer to Y than 1 /n ε , for any fixed ε > n ∈ N except at most finitely many.For any z ∈ Y , we have V N ( z, ε ) = O (cid:0) N ε − η (cid:1) for some fixed η > ε > ϑ = η − ε >
0. With theestimate of Eq. (3.1) in the proof of Theorem 3.5, we again obtain D N · V N ( z, ε ) = O (1)as N → ∞ , which establishes a sufficient criterion for the claimed convergence.Indeed, let 0 < δ < δ ′ be arbitrary, and let 1 δ denote the characteristic function of theset R \ (cid:0) Y + ( − δ, δ ) (cid:1) . Obviously, for any such δ , the function f ( δ ) := f · δ is both S-almostperiodic and locally Riemann-integrable on R . For a.e. x ∈ R , we thus get1 N N − X n =0 f ( δ ) ( α n x ) N →∞ −−−−→ M (cid:0) f ( δ ) (cid:1) δ ց −−−→ M ( f )by a combination of our previous arguments. Since the average of f − f ( δ ) along the exponentialsequence is controlled by the above mentioned estimate from Eq. (3.1), our claim follows. (cid:3) Note that our assumption on η achieves the analogue of the comment made in Remark 3.6.Note also that Remark 3.7 has an obvious extension to this more general situation. Indeed,one can once again replace the differentiability condition by the corresponding behaviour ofthe total variation in the vicinity of the ‘bad’ points.5. Further directions and extensions
Our exposition so far used complex-valued almost periodic functions over R , mainly for easeof presentation. More generally, one is interested in vector-valued functions, or in functionwith values in an arbitrary Banach space X , with norm | . | say. So, let f : R −−→ X be such afunction, and define k f k ∞ = sup x ∈ R | f ( x ) | . Then, the ε -almost periods of f are again definedas P ε := { t ∈ R : k f − T t f k ∞ < ε } , with (cid:0) T t f (cid:1) ( x ) = f ( x − t ) as before. Likewise, one can define trigonometric polynomials (or functions), by which one now meansany function T : R −−→ X of the form(5.1) Q m ( x ) = a + m X ℓ =1 e π i k ℓ x a ℓ for some m >
0, where { k , . . . , k ℓ } are distinct, non-zero real numbers and where the a ℓ arenow elements of X . When m = 0, the sum is meant to be empty and Q m is constant. Theanalogue of Fact 4.1 can now be stated as follows; see [14, Ch. VI] for details. Fact 5.1.
Let ( X , | . | ) be a Banach space. Then, for a continuous function f : R −−→ X , thefollowing properties are equivalent. (1) f is Bohr-almost periodic, i.e., P ε is relatively dense for any ε > ; (2) f is Bochner-almost periodic for k . k ∞ , i.e., the orbit { T t f : t ∈ R } is precompact inthe k . k ∞ -topology; (3) f is the limit of a sequence of trigonometric polynomials, with uniform convergenceof the sequence on R . (cid:3) There is no surprise up to this point, and we have gained rather little. To continue, weneed the notion of the mean of such a function f , and also some generalisation of the Fourierseries expansions. For this, we have to be able to (locally) integrate the function f . A naturalapproach is provided by Bochner’s integral [8], which can be viewed as an extension of theLebesgue integral to functions with values in a general Banach space; see [13, App. E] as wellas [40, Sec. V.5] for modern expositions.With this extension, most of our previous results remain true, with the only change thatthe coefficients a ℓ are now elements of X rather than complex numbers. For instance, onehas M ( Q m ) = a for the trigonometric polynomial of Eq. (5.1), and the analogue of Proposi-tion 4.2 holds without change. Now, also the consecutive steps have their natural analogues,and we obtain the following result. Theorem 5.2.
Let ( X , | . | ) be a Banach space, and let f : R −−→ X be Bohr-almost periodic.Then, for any fixed α ∈ R with | α | > , one has lim N →∞ N N − X n =0 f ( α n x ) = M ( f ) , which holds for a.e. x ∈ R . (cid:3) The extension to almost periodic functions in the Stepanov sense works in complete analogy,and we leave further steps in this direction to the reader.
Acknowledgements
MB would like to thank Jean-Pierre Conze, Michael Coons, Uwe Grimm and Nicolae Strun-garu for discussions and helpful comments. Financial support by the German Research Coun-cil (DFG) through CRC 701 is gratefully acknowledged.
VERAGING ALMOST PERIODIC FUNCTIONS 17
References [1] Apostol T.M.
An Introduction to Analytic Number Theory (Springer, New York, 1976).[2] Baake M. and G¨ahler F. Pair correlations of aperiodic inflation rules via renormalisation: Some interestingexamples,
Topol. Appl. (2016), 4–27; arXiv:1511.00885 .[3] Baake M. and Grimm U.
Aperiodic Order. Vol. 1: A Mathematical Invitation (Cambridge UniversityPress, Cambridge, 2013).[4] Baake M. and Haynes A. A measure theoretic result for approximation by Delone sets,
Preprint arXiv:1702.04839 .[5] Baake M. and Moody R.V. Weighted Dirac combs with pure point diffraction,
J. Reine und Angew. Math.(Crelle) (2004), 61–94; arXiv:math.MG/0203030 .[6] Besicovitch A.S.
Almost Periodic Functions , reprint (Dover, New York, 1954).[7] Besicovitch A. and Bohr H. Some remarks on generalisations of almost periodic functions,
Dan. Math.Fys. Medd. (1927), 1–31.[8] Bochner S. Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind, FundamentaMath. (1933), 262–276.[9] Bohr H. Almost Periodic Functions , reprint (Chelsea, New York, 1947).[10] Bufetov A.I. and Solomyak B. On the modulus of continuity for spectral measures in substitution dynamics,
Adv. Math. (2014), 84–129; arXiv:1305.7373 .[11] Bugeaud Y.
Distribution Modulo One and Diophantine Approximation , (Cambridge University Press,Cambridge, 2012).[12] Cigler J. Ziffernverteilung in ϑ -adischen Br¨uchen, Math. Z. (1964), 8–13.[13] Cohn D.L. Measure Theory , 2nd ed. (Birkh¨auser/Springer, New York, 2013).[14] Corduneanu C.
Almost Periodic Functions , 2nd English ed. (Chelsea, New York, 1989).[15] Cornfeld I.P., Fomin S.V. and Sinai Ya.G.
Ergodic Theory , SCSM 245 (Springer, New York, 1982).[16] de Bruijn N.G. and Post K.A. A remark on uniformly distributed sequences and Riemann integrability,
Nederl. Akad. Wetensch. Proc. Ser. A (1968), 149–150; now available as Indag. Math. (Proc.) (1968), 149–150.[17] Drmota M. and Tichy R.F. Sequences, Discrepancies and Applications , LNM 1651 (Springer, Berlin, 1997).[18] Fan A.-H., Saussol B. and Schmeling J. Products of non-stationary random matrices and multiperiodicequations of several scaling factors,
Pacific J. Math. (2004), 31–54; arXiv:math/0210347 .[19] Hardy G.H.
Divergent Series (Clarendon Press, Oxford, 1949).[20] Harman G.
Metric Number Theory (Oxford University Press, New York, 1998).[21] Hartinger J., Kainhofer R.F. and Tichy R.F. Quasi-Monte Carlo algorithms for unbounded, weightedintegration problems,
J. Complexity (2004), 654–668.[22] Kac M. On the distribution of values of sums of the type P f (2 k t ), Ann. Math. (1946), 33–49.[23] Kamarul Haili H. and Nair R. The discrepancy of some real sequences, Math. Scand. (2003), 268–274.[24] Katznelson Y. An Introduction to Harmonic Analysis , 3rd ed. (Cambridge University Press, Cambridge,2004).[25] Kuipers L. and Niederreiter H.
Uniform Distribution of Sequences , reprint (Dover, New York, 2006).[26] Leobacher G. and Pillichshammer F.
Introduction to Quasi-Monte Carlo Integration and Applications (Birkh¨auser, Basel, 2014).[27] Lesigne E. Loi des grands nombres pour des sommes de Riesz–Raikov multidimensionelles,
CompositioMath. (1998), 39–49. [28] Loomis L.H.
Introduction to Abstract Harmonic Analysis , reprint (Dover, New York, 2011).[29] Moody R.V. and Strungaru N. Almost periodic measures and their Fourier transforms. In
Aperiodic Order.Vol. 2: Crystallography and Almost Periodicity , eds. Baake M. and Grimm U. (Cambridge UniversityPress, Cambridge, 2017).[30] Parry W. On the β -expansion of real numbers, Acta Math. Acad. Sci. Hungar. (1960), 401–416.[31] Petit B. Le th´eor`eme limite central pour des sommes de Riesz–Raikov, Probab. Th. Rel. Fields (1992),407–438.[32] R´enyi A. Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. (1957), 477–493.[33] Rio E. Lois fortes des grands nombres presque sˆures pour les sommes de Riesz–Raikov, Probab. Th. Rel.Fields (2000), 342–348.[34] Sobol I.M. Calculation of improper integrals using uniformly distributed sequences,
Soviet Math. Dokl. (1973), 734–738.[35] Stepanoff W. ¨Uber einige Verallgemeinerungen der fastperiodischen Funktionen, Math. Ann. (1925),473–498.[36] Strungaru N. Almost periodic pure point measures. In Aperiodic Order. Vol. 2: Crystallography and AlmostPeriodicity , eds. Baake M. and Grimm U. (Cambridge University Press, Cambridge, 2017).[37] Takahashi S. On the distribution of values of the type P f ( q k t ), Tohoku Math. J. (1962), 233–243.[38] Walters P. An Introduction to Ergodic Theory , reprint (Springer, New York, 2000).[39] Weyl H. ¨Uber die Gleichverteilung von Zahlen mod. Eins,
Math. Annalen (1916), 313–352.[40] Yoshida K. Functional Analysis , SCSM 123, 6th ed. (Springer, Berlin, 1980).
Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld,Postfach 100131, 33501 Bielefeld, Germany
E-mail address : [email protected] Department of Mathematics, University of Houston,3551 Cullen Blvd., Houston, TX 77204-3008, USA
E-mail address : [email protected] Fakult¨at f¨ur Mathematik, Universit¨at Jena,Ernst-Abbe-Platz 2, 07743 Jena, Germany
E-mail address ::