aa r X i v : . [ m a t h . N T ] J un XXXX , 1–20 © De Gruyter YYYY
The hybrid spectral test
Peter Hellekalek
Abstract.
The starting point of this paper is the interplay between the construction principleof a sequence and the characters of the compact abelian group that underlies the construction.In case of the Halton sequence in base b = ( b , . . . , b s ) in the s -dimensional unit cube [ , ) s ,which is an important type of a digital sequence, this kind of duality principle leads to theso-called b -adic function system and provides the basis for the b -adic method, which wepresent in connection with hybrid sequences. This method employs structural properties of thecompact group of b -adic integers as well as b -adic arithmetic to derive tools for the analysisof the uniform distribution of sequences in [ , ) s .We first clarify the point which function systems are needed to analyze digital sequences.Then, we present the hybrid spectral test in terms of trigonometric-, Walsh-, and b -adic func-tions. Various notions of diaphony as well as many figures of merit for rank-1 quadraturerules in Quasi-Monte Carlo integration and for certain linear types of pseudo-random numbergenerators are included in this measure of uniform distribution. Further, discrepancy may beapproximated arbitrarily close by suitable versions of the spectral test. Keywords.
Uniform distribution of sequences, discrepancy, diaphony, Fourier series, Walshseries, spectral test, pseudorandom number generation, hybrid sequences.
AMS classification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
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This paper is about certain mathematical concepts to analyze the uniform distribu-tion behaviour of sequences on the s -dimensional torus [ , ) s . We discuss qualitativeaspects, by which we understand the study of properties of an infinite sequence thatinduce its uniform distribution in [ , ) s , and also quantitative aspects, i.e. the questionhow to measure the uniform distribution of a finite or infinite sequence on the s -torus.A general setting to generate a (finite or infinite) sequence ω = ( x n ) n ≥ in someoutput space O , like O = [ , ) s , is the following. We consider a nonempty set S , P. Hellekalek the so-called state space, and a map T : S → S , the state update transformation.We employ T to generate a (finite or infinite) sequence of states ( s n ) n ≥ in S . Thissequence is then mapped to a sequence ω = ( x n ) n ≥ in O by an output map ϕ : S →O , by letting x n = ϕ ( s n ) , n ≥ ( s n ) n ≥ is constructed byiterating T . We start with some initial state s and put s n = T n ( s ) , n ≥
0. Illustrativeexamples are linear or inversive congruential pseudorandom number generators (see[29]), the well known ( nα ) n ≥ sequences (see [24]), or the block cipher AES in OutputFeedback mode (see [2, p. 28]).In some other cases, we employ a ring ( R, + , · ) with unity 1 and a function ψ : R → S to map the sequence ( n ) n ≥ (the so-called “counter”) first to a sequence ofstates ( ψ ( n )) n ≥ in S . Then, this sequence is “encrypted” by T to give the sequence ( T ◦ ψ ( n )) n ≥ in S . Finally, this sequence of states is mapped into the output space O . This gives ω = ( ϕ ◦ T ◦ ψ ( n )) n ≥ . Examples of such constructions are explicitinversive congruential pseudorandom number generators (see [11, 30, 31, 36]), certaindigital sequences (see [3]), or AES in Counter mode (see [2, p. 28]). In these examples,we let R = Z , the integral domain of integers.In all of the cases exhibited above, we employ some arithmetical operation likeaddition on the state space S . If S is a compact abelian group with respect to the chosenoperation, then we have the arsenal of abstract harmonic analysis at our disposition(see [22]). The choice of the compact group S determines which function systemis suitable for the analysis of the equidistribution behavior of the sequence ω in O ,because the group operation on S is intrinsically related to the dual group ˆ S of S .An example of such a suitable match between sequences and function systems basedon this duality principle is given by Kronecker sequences or, in the discrete version,good lattice points, and the trigonometric functions. Here, S = O = [ , ) s , and theconstruction method uses addition modulo one on the s -torus [ , ) s , T ( s ) = s + α ,with α ∈ R s (see [4], [29, Ch. 5] and [34]). The dual of the compact group S may beinterpreted as the trigonometric function system. For the background of these grouprotations in ergodic theory, we refer to the monographs [32, 37].A second example of this duality principle is given by digital nets and sequencesand the Walsh functions. Here, addition without carry of digit vectors comes into play(see [29, Ch. 4] and [3]). For example, for nets and sequences in base 2, the underlyinggroup S is the compact group F ∞ and its dual group is the Walsh function system inbase 2. The same relation holds for general integer bases b ≥ b =( b , . . . , b s ) on the s -torus [ , ) s , is generated by addition with carry of digit vectors.For this reason, it makes sense to choose as the underlying group S the compact groupof b -adic integers Z b × · · · × Z b s . Considering the dual group then leads to the b -adicfunction system, which is the main tool of the b -adic method introduced in [14, 15, 17].It will be discussed in Section 3.The general duality principle presented before also accommodates for hybrid se- he hybrid spectral test quences , which are sequences ω of points in [ , ) s where certain coordinates of thepoints stem from one lower-dimensional sequence ω , with state space S , and the re-maining coordinates from a second lower-dimensional sequence ω , with state space S . The duality principle leads us to consider hybrid function systems , which arisefrom the product group ˆ S × ˆ S . Of course, this idea may be generalized to mixingmore than two subsequences into a hybrid sequence.This paper is structured as follows. In Section 2 we consider the question whichfunction systems will suffice to analyze digital sequences. In Section 3 we introducethe necessary notation and in Section 4 we define the hybrid spectral test and show thatit is a measure of the uniform distribution of a sequence in [ , ) s . Section 5 deals withspecial cases of the spectral test, like diaphonies, the spectral test for pseudorandomnumber generators, various figures of merit for integer lattice points, like they appearin the context of good lattice points and rank-1 lattice rules. Finally, we show thatthe extreme as well as the star discrepancy can be approximated arbitrarily close byspecial cases of the hybrid spectral test. Digital sequences on the s -torus are sequences that arise from operations with digitvectors in some given integer bases b i ≥
2, 1 ≤ i ≤ s . For the sake of simplicity, werestrict the following discussion to the one-dimensional case.As we have seen in Section 1, and as the block ciphers IDEA [25] and AES [2]illustrate in cryptography, two types of addition of digit vectors are in use in applica-tions, addition without carry and addition with carry. By the duality principle, the firstaddition leads to Walsh functions and the second type to b -adic functions as the propertools for the analysis of such sequences.Are these two types of function systems sufficient to study digital sequences? Inalgebraic terms, are there any other possibilities to add digit vectors than addition withor without carry? If yes, then this would lead to additional types of groups and dualgroups and, hence, to additional function systems. If not, then the Walsh functions andthe b -adic functions suffice to analyze digital sequences.For details, in particular for the proofs in the following considerations and a refine-ment using compositions of positive integers instead of partitions and automorphismsof certain groups to define additions, we refer the reader to [16].Let b ≥ A b = { , , . . . , b − } denote the set of b -arydigits. For m ∈ N , m ≥
2, let A mb stand for the m -fold cartesian product of the set A b with itself. We study the following question: What are the binary operations “ + ” onthe set A mb of digit vectors of length m such that the pair ( A mb , +) is an abelian group?In this paper, when we speak of an “addition on A mb ”, we mean a binary operation“ + ” on the set A mb of digit vectors in base b such that the pair ( A mb , +) is an abeliangroup. The reader should note that the term “binary” has two different meanings here,which will become clear from the context. A binary operation on a set G is a map from P. Hellekalek the cartesian product G × G into G . Referring to the representation of real numbers inbase b =
2, the elements of the set A m are called binary vectors, and for m = b = n ∈ N , n ≥
2, let Z /n Z denote the additive group of residue classes modulo n .We identify this cyclic group with the set of integers { , , . . . , n − } equipped withaddition modulo n . Example 2.1.
We identify A with Z / Z . For x , y ∈ A m , x = ( x , . . . , x m − ) and y = ( y , . . . , y m − ) , we define x + y = ( x ⊕ y , . . . , x m − ⊕ y m − ) , where ‘ ⊕ ’ denotes addition on Z / Z , 0 ⊕ = ⊕ =
0, and 0 ⊕ = ⊕ = ( A m , +) is an abelian group. In fact, it is isomorphic to the product group ( Z / Z ) m . We call this binary operation addition without carry , or XOR -addition ofdigit vectors.Every nonnegative integer k , 0 ≤ k < m , has a unique representation in base 2 ofthe form k = k + k + · · · + k m − m − with digits k j ∈ A , 0 ≤ j ≤ m − Example 2.2.
We identify A m with the group Z / m Z as follows. For x ∈ A m , x =( x , . . . , x m − ) , we define the map int : A m → Z / m Z ,int ( x ) = x + x + · · · + x m − m − . Further, let dig : Z / m Z → A m ,dig ( k ) = ( k , k , . . . , k m − ) , where k = k + k + · · · + k m − m − is the representation of k in base 2. Finally,for x , y ∈ A m , we define x + y = dig ( int ( x ) + int ( y ) ( mod 2 m )) . With this binary operation, the pair ( A m , +) is an abelian group. Clearly, it is iso-morphic to the additive group Z / m Z . We call this type of binary operation additionwith carry or integer addition of digit vectors.For m ≥
2, our two examples of addition act on non-isomorphic groups, because Z / m Z is cyclic and ( Z / Z ) m is not. Apart from these two examples, are there anyother possibilities to define addition on the set A m ? From the Fundamental Theoremfor Finite Abelian Groups (see, for example, [21, Sec. 10]) we obtain the followinglemma. In this context, a partition of a positive integer m is a finite sequence ( t i ) ri = , r ∈ N , of positive integers t i with the two properties (i) t ≥ t ≥ · · · ≥ t r , and (ii) t + t + · · · + t r = m . he hybrid spectral test Lemma 2.3.
The non-isomorphic groups of order m , m ∈ N , are given by the productgroups ( Z / t Z ) × ( Z / t Z ) × · · · × ( Z / t r Z ) , where ( t i ) ri = is a partition of m . Hence, in view of Lemma 2.3, an addition on the set A m is defined if we choose apartition m = t + t + · · · + t r of m and put ( A m , +) ∼ = ( Z / t Z ) × ( Z / t Z ) × · · · × ( Z / t r Z ) . (2.1)Here, the symbol “ ∼ = ” denotes that the two groups are isomorphic. From the structureof the factors in (2.1) we obtain the following information. Theorem 2.4.
The only two types of binary operations on (sub)vectors of digits thatmay appear in the group law of the abelian group ( A m , +) are the following: • addition given by finite product groups of the form ( Z / Z ) ×· · · × ( Z / Z ) , whichis what we have called addition without carry, or • addition in groups of residues of the form Z / t Z , t ≥ , which we have calledaddition with carry. Theorem 2.4 directly generalizes from base 2 to arbitrary prime bases p . In the caseof a composite base b , we also have the equivalent of ( 2.1), as well as some additionaldirect products arising from the factorization of b into prime powers. Even in the lattercases, only two types of addition of (sub)vectors of digits appear, addition with orwithout carry. Remark 2.5.
We have seen that there exist only two types of addition for digit vec-tors, while there exist many different variants of this binary operation, for exampleby changing the underlying partition sequence. As a consequence, from the dualityprinciple only two types of function systems arise, the Walsh functions and the b -adicfunctions. For those digits that are added without carry, the Walsh system applies, andfor those digits that are added with carry, the b -adic system of functions is appropriate.These two function systems cover all possible cases. Throughout this paper, b denotes a positive integer, b ≥
2, and b = ( b , . . . , b s ) standsfor a vector of not necessarily distinct integers b i ≥
2, 1 ≤ i ≤ s . N represents thepositive integers, and we put N = N ∪ { } . The underlying space is the s -dimensional torus R s / Z s , which will be identifiedwith the half-open interval [ , ) s . Haar measure on the s -torus [ , ) s will be denotedby λ s . We put e ( y ) = e π i y for y ∈ R , where i is the imaginary unit. P. Hellekalek
We will use the standard convention that empty sums have the value 0 and emptyproducts value 1.For a nonnegative integer k , let k = P j ≥ k j b j , k j ∈ { , , . . . , b − } , be theunique b -adic representation of k in base b . With the exception of at most finitelymany indices j , the digits k j are equal to 0.Every real number x ∈ [ , ) has a representation in base b of the form x = P j ≥ x j b − j − , with digits x j ∈ { , , . . . , b − } . If x is a b -adic rational , whichmeans that x = ab − g , a and g integers, 0 ≤ a < b g , g ∈ N , and if x =
0, then thereexist two such representations.The b -adic representation of x is uniquely determined under the condition that x j = b − j . In the following, we will call this particular representationthe regular ( b -adic) representation of x .Let Z b denote the compact group of the b -adic integers. We refer the reader to [22,28] for details. An element z of Z b will be written as a formal sum z = P j ≥ z j b j , with digits z j ∈ { , , . . . , b − } . The set Z of integers is embedded in Z b . If z ∈ N ,then at most finitely many digits z j are different from 0. If z ∈ Z , z <
0, then at mostfinitely many digits z j are different from b −
1. In particular, − = P j ≥ ( b − ) b j . We recall the following concepts from [15, 17, 20].
Definition 3.1.
The map ϕ b : Z b → [ , ) , given by ϕ b ( P j ≥ z j b j ) = P j ≥ z j b − j − ( mod 1 ) , will be called the b -adic Monna map .The restriction of ϕ b to N is often called the radical-inverse function in base b . TheMonna map is surjective, but not injective. It may be inverted in the following sense. Definition 3.2.
We define the pseudoinverse ϕ + b of the b -adic Monna map ϕ b by ϕ + b : [ , ) → Z b , ϕ + b ( X j ≥ x j b − j − ) = X j ≥ x j b j , where P j ≥ x j b − j − stands for the regular b -adic representation of the element x ∈ [ , ) .The image of [ , ) under ϕ + b is the set Z b \ ( − N ) . Furthermore, ϕ b ◦ ϕ + b is the iden-tity map on [ , ) , and ϕ + b ◦ ϕ b the identity on N ⊂ Z b . In general, z = ϕ + b ( ϕ b ( z )) ,for z ∈ Z b . For example, if z = −
1, then ϕ + b ( ϕ b ( − )) = ϕ + b ( ) = = − Z b can be written in the form ˆ Z b = { χ k : k ∈ N } , where χ k : Z b → { c ∈ C : | c | = } , χ k ( P j ≥ z j b j ) = e ( ϕ b ( k )( z + z b + · · · )) . We note that χ k depends only on a finite number of digits of z and, hence,this function is well defined.As in [15], we employ the function ϕ + b to lift the characters χ k to the torus. Definition 3.3.
For k ∈ N , let γ k : [ , ) → { c ∈ C : | c | = } , γ k ( x ) = χ k ( ϕ + b ( x )) ,denote the k th b -adic function. We put G b = { γ k : k ∈ N } and call it the b -adicfunction system on [ , ) . he hybrid spectral test b = ( b , . . . , b s ) be a vector of not necessarily distinct integers b i ≥
2, let x = ( x , . . . , x s ) ∈ [ , ) s , let z = ( z , . . . , z s ) denote an element of the compactproduct group Z b = Z b × · · · × Z b s of b -adic integers, and let k = ( k , . . . , k s ) ∈ N s .We define ϕ b ( z ) = ( ϕ b ( z ) , . . . , ϕ b s ( z s )) , and ϕ + b ( x ) = ( ϕ + b ( x ) , . . . , ϕ + b s ( x s )) .Let χ k ( z ) = Q si = χ k i ( z i ) , where χ k i ∈ ˆ Z b i , and define γ k ( x ) = Q si = γ k i ( x i ) ,where γ k i ∈ G b i , 1 ≤ i ≤ s . Then γ k = χ k ◦ ϕ + b . Let G ( s ) b = { γ k : k ∈ N s } denotethe b -adic function system in dimension s .The dual group ˆ Z b is an orthonormal basis of the Hilbert space L ( Z b ) . A ratherelementary proof of this result is given in [20, Theorem 2.12].For the Walsh functions defined below, we refer the reader to [3, 10, 12] for elemen-tary properties of these functions and to [33] for the background in harmonic analysis. Definition 3.4.
For k ∈ N , k = P j ≥ k j b j , and x ∈ [ , ) , with regular b -adicrepresentation x = P j ≥ x j b − j − , the k th Walsh function in base b is defined by w k ( x ) = e (( P j ≥ k j x j ) /b ) . For k ∈ N s , k = ( k , . . . , k s ) , and x ∈ [ , ) s , x =( x , . . . , x s ) , we define the k th Walsh function w k in base b = ( b , . . . , b s ) on [ , ) s as the following product: w k ( x ) = Q si = w k i ( x i ) , where w k i denotes the k i th Walshfunction in base b i , 1 ≤ i ≤ s . The Walsh function system in base b in dimension s isdenoted by W ( s ) b = { w k : k ∈ N s } .The trigonometric function system defined below is the classical function system inthe theory of uniform distribution of sequences (see the monograph [24]). Definition 3.5.
Let k ∈ Z . The k th trigonometric function e k is defined as e k : [ , ) → C , e k ( x ) = e ( kx ) . For k = ( k , . . . , k s ) ∈ Z s , the k th trigonometricfunction e k is defined as e k : [ , ) s → C , e k ( x ) = Q si = e ( k i x i ) , x = ( x , . . . , x s ) ∈ [ , ) s . The trigonometric function system in dimension s is denoted by T ( s ) = { e k : k ∈ Z s } .The following presentation complements the concepts discussed in [17]. As willbecome clear, any finite number of factors can be accomodated. For given dimensions s , s and s , with s i ∈ N , not all equal to 0, put s = s + s + s and write a point y ∈ R s in the form y = ( y ( ) , y ( ) , y ( ) ) with components y ( j ) ∈ R s j , j = , ,
3. Letus fix two vectors of bases b ( ) = ( b , . . . , b s ) , and b ( ) = ( b s + , . . . , b s + s ) withnot necessarily distinct integers b i ≥
2. Let k = ( k ( ) , k ( ) , k ( ) ) , with components k ( ) ∈ N s , k ( ) ∈ N s , and k ( ) ∈ Z s . The tensor product ξ k = w k ( ) ⊗ γ k ( ) ⊗ e k ( ) ,where w k ( ) ∈ W ( s ) b ( ) , γ k ( ) ∈ G ( s ) b ( ) , and e k ( ) ∈ T ( s ) , defines a function ξ k on the s -dimensional unit cube, ξ k : [ , ) s → C , ξ k ( x ) = w k ( ) ( x ( ) ) γ k ( ) ( x ( ) ) e k ( ) ( x ( ) ) , where x = ( x ( ) , x ( ) , x ( ) ) ∈ [ , ) s . P. Hellekalek
Definition 3.6.
Let s , s , s ∈ N , not all s i equal to 0, and put s = s + s + s . Thefamily of functions W ( s ) b ( ) ⊗ G ( s ) b ( ) ⊗ T ( s ) = { ξ k , k = ( k ( ) , k ( ) , k ( ) ) ∈ N s × N s × Z s } , is called a hybrid function system on [ , ) s . Remark 3.7.
It follows from [17, Theorem 1 and Corollary 4] and the techniques ex-hibited in [20] that such hybrid function systems are an orthonormal basis of L ([ , ) s ) . Remark 4.1.
All of the following results remain valid if we change the order of thefactors in the hybrid function system, as it will become apparent from the proofs below.In particular, out of the given s coordinates, we may select arbitrary s coordinates andassign to them the Walsh system W ( s ) b ( ) in some base b ( ) , treat s of the remaining s − s coordinates with a b ( ) -adic system G ( s ) b ( ) , and use the system T ( s ) for the final s coordinates. Definition 4.2.
Let s ∈ N . By an s -dimensional index set L we understand one of theadditive semigroups ( Z s , +) , ( N s , +) , and ( N s , +) , or finite direct products of thesesemigroups such that the dimensions of the factors add up to s .Let L ∗ denote the index set L \ { } .Examples of s -dimensional index sets are direct products of the form N s × N s × Z s , where s = s + s + s , with s , s , s ∈ N , not all s i equal to 0, as they appearin hybrid function systems.If ω = ( x n ) n ≥ is a -possibly finite- sequence in [ , ) s with at least N elements,and if f : [ , ) s → C , we define S N ( f, ω ) = N N − X n = f ( x n ) . Definition 4.3.
Let L be an s -dimensional index set. A subclass F = { ξ k : k ∈ L } of the class of Riemann integrable functions on [ , ) s is called a uniform distributiondetermining (u.d.d.) function system on [ , ) s if the functions ξ k are normalized in thesense that || ξ k || ∞ = sup {| ξ k ( x ) | : x ∈ [ , ) s } ≤ k , and R [ , ) s ξ k dλ s = k ∈ L ∗ , and if, for any sequence ω in [ , ) s , the property ∀ k ∈ L ∗ : lim N →∞ S N ( ξ k , ω ) = ω in [ , ) s . he hybrid spectral test [ , ) s are the hybrid function systems W ( s ) b ( ) ⊗ G ( s ) b ( ) ⊗ T ( s ) . This follows from the hybrid Weyl Criterion [17, Theorem 1],whose proof is easily adapted to provide for the non-prime bases b i we allow here inthe factor G ( s ) b ( ) . Definition 4.4.
Let || · || be an arbitrary norm on R s and let L be an s -dimensionalindex set. We call a real-valued function ρ a weight function on L if, for all k ∈ L , ρ ( k ) >
0, and if for all ǫ >
0, there exists a positive real number K = K ( ǫ ) suchthat ρ ( k ) < ǫ for all k ∈ L with || k || > K .With a u.d.d. function system F = { ξ k : k ∈ L } on [ , ) s and a weight function ρ on L we may associate the array of weighted functions ( ρ ( k ) ξ k ) k ∈ L . The operator S N ( · , ω ) is linear, hence we may write S N (( ρ ( k ) ξ k ) k ∈ L , ω ) = ( ρ ( k ) S N ( ξ k , ω )) k ∈ L . Definition 4.5.
Let F = { ξ k : k ∈ L } be a u.d.d. function system on [ , ) s , and let ρ be a weight function on L . For a given sequence ω = ( x n ) n ≥ in [ , ) s , the spectraltest σ N ( ω ) of the first N elements of ω , with respect to F and ρ , is defined as σ N ( ω ) = || ( ρ ( k )) k ∈ L ∗ || − ∞ || ( ρ ( k ) S N ( ξ k , ω )) k ∈ L ∗ || ∞ = sup k ∈ L ∗ { ρ ( k ) } − sup k ∈ L ∗ { ρ ( k ) | S N ( ξ k , ω ) |} . Let α > ρ fulfills the additionalcondition X k ∈ L ρ ( k ) α < ∞ , then the L α - diaphony F ( α ) N ( ω ) of the first N elements of ω , with respect to F and ρ ,is defined as F ( α ) N ( ω ) = || ( ρ ( k )) k ∈ L ∗ || − α || ( ρ ( k ) S N ( ξ k , ω )) k ∈ L ∗ || α = X k ∈ L ∗ ρ ( k ) α ! − /α X k ∈ L ∗ ρ ( k ) α | S N ( ξ k , ω ) | α ! /α . We note that the fact | S N ( ξ k , ω ) | ≤ ≤ σ N ( ω ) ≤
1, and 0 ≤ F ( α ) N ( ω ) ≤ Theorem 4.6.
Let F , ρ and σ N ( ω ) be as in Definition 4.5. Then P. Hellekalek (i)
The quantity σ N ( ω ) is a maximum. (ii) The sequence ω is uniformly distributed modulo one if and only if lim N →∞ σ N ( ω ) = . Proof.
The proof generalizes the arguments in [12, Sec. 5.2]. For an arbitrary positiveinteger K , let A N,K = sup { ρ ( k ) | S N ( ξ k , ω ) | : 0 < || k || ≤ K } , and B N,K = sup { ρ ( k ) | S N ( ξ k , ω ) | : || k || > K } . Clearly, σ N ( ω ) = max { A N,K , B
N,K } . (4.1)We have σ N ( ω ) >
0. Otherwise, all terms S N ( ξ k , ω ) , k ∈ L ∗ , would be equal to zero,which is impossible. Hence, there exists δ > δ < σ N ( ω ) , and an index K = K ( δ ) ∈ N such that for all k with || k || > K we have ρ ( k ) < δ . This implies B N,K ≤ sup { ρ ( k ) : || k || > K } ≤ δ < σ N ( ω ) . Hence, σ N ( ω ) = A N,K . We observe that the set { k : || k || ≤ K } is finite. In thiscontext, we recall that all norms on R s are equivalent. This proves (i).In order to prove (ii), suppose first that lim N →∞ σ N ( ω ) =
0. This implies thatlim N →∞ S N ( ξ k , ω ) = k ∈ L ∗ . The class F is u.d.d., hence ω is uniformlydistributed in [ , ) s .To prove the converse, assume that ω is uniformly distributed in [ , ) s . For any ǫ >
0, there exists a positive integer K = K ( ǫ ) such that ρ ( k ) < ǫ for all k with || k || > K . As in the proof of part (i), this gives B N,K ≤ ǫ , and, due to (4.1), σ N ( ω ) ≤ A N,K + ǫ. The number A N,K is a maximum and the class F is u.d.d.. This implies the existenceof N = N ( ǫ ) ∈ N with the property ∀ N ≥ N ( ǫ ) : A N,K = max { ρ ( k ) | S N ( ξ k , ω ) | : 0 < || k || ≤ K } < ǫ. We deduce the relation ∀ N ≥ N ( ǫ ) : σ N ( ω ) < ǫ. This proves the theorem. he hybrid spectral test Corollary 4.7.
Let F , ρ and σ N ( ω ) be as in Definition 4.5 and let K denote an arbi-trary positive integer. Then we have the following inequality of Erdös-Turán-Koksmafor the spectral test: σ N ( ω ) ≤ max n max < || k ||≤ K { ρ ( k ) | S N ( ξ k , ω ) |} , sup || k || >K { ρ ( K ) } o . Proof.
This follows directly from (4.1).
Theorem 4.8.
Let F , ρ and F ( α ) N ( ω ) be as in Definition 4.5 and suppose that X k ∈ L ∗ ρ ( k ) α < ∞ . Then sequence ω is uniformly distributed in [ , ) s if and only if lim N →∞ F ( α ) N ( ω ) = . Proof.
We adapt the proof of [17, Theorem 2] and the splitting technique used in theproof of Theorem 4.6 to the case of diaphony.
Definition 4.5 generalizes various known notions of the spectral test and of diaphony.The spectral test has its origin in pseudorandom number generation.It measures the“coarseness” of lattices that can be associated with certain linear types of generators.We refer to [23, Ch. 3.3.4.]) for a seminal discussion, and to the surveys [26, 29] aswell as to [5].We recall some notions from the theory of lattices (for details see [29, 34]). An s -dimensional lattice is a discrete subset of R s which is closed under addition andsubtraction. For any s -dimensional lattice L , there exists a lattice basis, by which weunderstand s independent vectors g , . . . , g s in R s such that L = ( s X i = t i g i : t i ∈ Z ) . Lattice bases are not unique.A point g ∈ L \ { } is called a primitive point of L if the line segment joining theorigin and g does not contain any other point of L .The dual lattice L ⊥ of L is defined as L ⊥ = { z ∈ R s : z · x ∈ Z , for all x ∈ L } , P. Hellekalek where z · x denotes the usual inner product.An s -dimensional integration lattice is an s -dimensional lattice that contains Z s asa sublattice. An s -dimensional N -point lattice rule is an s -dimensional integrationlattice L of the form L = N − [ n = ( x n + Z s ) , (5.1)where x , . . . , x N − are the N distinct points of L that belong to [ , ) s .The spectral test for lattices is defined as follows. Definition 5.1.
Let L be an s -dimensional lattice in R s . We will call a family H C ofparallel hyperplanes H c , c ∈ C , in R s a cover of L if (i) L ⊆ S c ∈ C H c , and (ii) C isthe smallest set (in the sense of set-inclusion) with this property.Let H C denote an arbitrary cover of L in R s . The spacing d ( H C ) of H C willdenote the minimal distance between adjacent hyperplanes in this family, d ( H C ) = inf { d ( H c , H d ) : c = d, c, d ∈ C } . We define the spectral test of L as the number σ ( L ) : = sup { d ( H C ) : H C is a cover of L } . The following theorem is well known in the theory of pseudorandom number gen-eration (see [6, 12, 23, 27]). Computational aspects are discussed in [1].
Theorem 5.2.
Let L be an s -dimensional lattice. Then σ ( L ) = / min {|| g || : g ∈ L ⊥ , g primitive } , where || g || = ( g + · · · + g s ) / is the Euclidean norm on R s . In the field of quasi-Monte Carlo integration, similar concepts have been developedwithin the context of good lattice points . Definition 5.3.
For a given integer N ≥ s ≥
2, we call aninteger point a = ( a , . . . , a s ) ∈ Z s , gcd ( a i , N ) =
1, 1 ≤ i ≤ s , a good lattice point modulo N if the finite sequence ω a = (cid:16)n nN a o(cid:17) N − n = (5.2)is “very uniformly” distributed in [ , ) s in the sense that its discrepancy is low.The conditions gcd ( a i , N ) =
1, 1 ≤ i ≤ s , ensure that all points in ω a and in itsprojections to lower dimensions are distinct.We may view ω a as the node set of an s -dimensional N -point lattice rule. Let L ( ω a ) denote the integration lattice associated with ω a defined by (5.1). The dual lattice isgiven by L ( ω a ) ⊥ = { k ∈ Z s : k · a ≡ ( mod N ) } . he hybrid spectral test a ∈ Z s subject to the conditions of Definition 5.3, we mayassociate several figures of merit. In view of the spectral test for lattices, define σ ( a , N ) = σ ( L ( ω a )) . Then σ ( a , N ) = / min {|| k || : k ∈ Z s \ { } , k · a ≡ ( mod N ) } . For a real number α >
1, define P α ( a , N ) = X k = · a ≡ ( mod N ) r ( k ) α , where summation is over integer vectors k , and r ( k ) = s Y i = max { , | k i |} , k = ( k , . . . , k s ) ∈ Z s . Another important figure of merit is the
Babenko-Zaremba index κ ( a , N ) = / min { r ( k ) : k ∈ Z s \ { } , k · a ≡ ( mod N ) } , see the monographs [29, 34].The three quantities σ ( a , N ) , P α ( a , N ) , and κ ( a , N ) are special cases of the spec-tral test introduced in Definition 4.5 applied to the sequence ω a . In order to establishthis connection, we employ the following result of Sloan and Kachoyan [35] (for theproof see [29, Lemma 5.21] and [34, Lemma 2.7]): Lemma 5.4.
Let ω = ( x n ) N − n = be the sequence of nodes of an s -dimensional N -pointlattice rule L . Then S N ( e k , ω ) = ( , if k ∈ L ⊥ , , if k L ⊥ . Example 5.5.
Let F = T ( s ) (hence L = Z s ), and put ω a = ( { ( n/N ) a } ) N − n = , where a ∈ Z s is subject to the conditions in Definition 5.3. Then,(i) for the choice ρ ( k ) = || k || − for k = , the hybrid spectral test σ N ( ω a ) intro-duced in Definition 4.5, with respect to F and ρ , is equal to the classical spectraltest of Definition 5.1 applied to the integration lattice L ( ω a ) .(ii) for ρ ( k ) = r ( k ) − for k = , we obtain σ N ( ω a ) = κ ( a , N ) , the Babenko-Zaremba index defined in (5.1).(iii) for ρ ( k ) = r ( k ) − for k = , and α >
1, we get X k = r ( k ) − α /α F ( α ) N ( ω a ) = ( P α ( a , N )) /α , for the L α -diaphony of the sequence ω a .4 P. Hellekalek
Example 5.6.
By obvious choices of F and ρ and with α =
2, we obtain the classicaldiaphony of Zinterhof [38], the dyadic (Walsh) diaphony of Hellekalek and Leeb [19],the b -adic (Walsh) diaphony versions of Grozdanov et al. [7, 8, 9], the p -adic diaphonyof Hellekalek [15], and the more general notion of hybrid diaphony that was introducedin [17]. Example 5.7.
By obvious choices for F and ρ , we obtain the Walsh spectral test ofHellekalek [13]. The hybrid spectral test introduced in Definition 4.5 does not include the extremediscrepancy and the star discrepancy, but we may approximate these two measures ofuniform distribution arbitrarily close by suitable versions of the hybrid spectral test.We recall the definition of discrepancy. Let J denote the class of all subintervalsof [ , ) s of the form Q si = [ u i , v i ) , 0 ≤ u i < v i ≤
1, 1 ≤ i ≤ s , and let J ∗ denotethe subclass of J of intervals of the type Q si = [ , v i ) anchored at the origin. Theextreme discrepancy and the star discrepancy of a sequence are defined as follows (see[24, 29]). Definition 5.8.
Let ω = ( x n ) n ≥ be a sequence in [ , ) s . The (extreme) discrepancy D N ( ω ) of the first N elements of ω is defined as D N ( ω ) = sup J ∈J | S N ( J − λ s ( J ) , ω ) | . The star discrepancy D ∗ N ( ω ) of the first N elements of ω is defined as D ∗ N ( ω ) = sup J ∈J ∗ | S N ( J − λ s ( J ) , ω ) | . We first approximate D N ( ω ) and D ∗ N ( ω ) by discrete discrepancies. Definition 5.9.
Let b = ( b , . . . , b s ) , with not necessarily distinct integers b i ≥ b -adic interval in the resolution class defined by g = ( g , . . . , g s ) ∈ N s (or withresolution g ) is a subinterval of [ , ) s of the form s Y i = h a i b − g i i , d i b − g i i (cid:17) , ≤ a i < d i ≤ b g i i , a i , d i ∈ N , ≤ i ≤ s . We denote the class of all b -adic intervals with resolution g by J b , g . The subclass ofthose b -adic intervals anchored at the origin will be denoted by J ∗ b , g . Further, let J b = [ g ∈ N s J b , g he hybrid spectral test b -adic intervals in [ , ) s and put J ∗ b = [ g ∈ N s J ∗ b , g . For a given resolution g ∈ N s , we define the domains D b ( g ) = { k = ( k , . . . , k s ) ∈ N s : 0 ≤ k i < b g i i , ≤ i ≤ s } , D ∗ b ( g ) = D b ( g ) \ { } . and ∇ b ( g ) = { k = ( k , . . . , k s ) ∈ N s : 1 ≤ k i ≤ b g i i , ≤ i ≤ s } . We note that D b ( ) = { } and ∇ b ( ) = ∅ . Further, we observe that we may writethe intervals in J b , g in the form J b , g = { I a , d ; g : ( a , d ) ∈ D b ( g ) × ∇ b ( g ) } , where I a , d ; g = Q si = [ ϕ b i ( a i ) , ϕ b i ( d i )) , and a = ( a , . . . , a s ) , and d = ( d , . . . , d s ) .The intervals in J ∗ b , g are of the form I , d ; g , with d ∈ ∇ b ( g ) . Definition 5.10.
Let ω = ( x n ) n ≥ be a sequence in [ , ) s , let b = ( b , . . . , b s ) , withnot necessarily distinct integers b i ≥
2, and let g ∈ N s be a given resolution vector.The discrete (extreme) discrepancy in base b , for resolution g , of the first N elementsof ω is defined as D N ; b , g ( ω ) = max I ∈J b , g | S N ( I − λ s ( I ) , ω ) | . The discrete star discrepancy in base b , for resolution g , of the first N elements of ω is defined as D ∗ N ; b , g ( ω ) = max I ∈J ∗ b , g | S N ( I − λ s ( I ) , ω ) | . Theorem 5.11.
Let b = ( b , . . . , b s ) be a vector of s not necessarily distinct integers b i ≥ . Then, for all g = ( g , . . . , g s ) ∈ N s , D N ; b , g ( ω ) ≤ D N ( ω ) ≤ ǫ b ( g ) + D N ; b , g ( ω ) ,D ∗ N ; b , g ( ω ) ≤ D ∗ N ( ω ) ≤ ǫ ∗ b ( g ) + D ∗ N ; b , g , where the error terms ǫ b ( g ) and ǫ ∗ b ( g ) are given by ǫ b ( g ) = − s Y i = ( − b − g i i ) , ǫ ∗ b ( g ) = − s Y i = ( − b − g i i ) . P. Hellekalek
Proof.
The inequalities D N ; b , g ( ω ) ≤ D N ( ω ) and D ∗ N ; b , g ( ω ) ≤ D ∗ N ( ω ) are trivial.In order to show D N ( ω ) ≤ ǫ b ( g ) + D N ; b , g ( ω ) , we proceed as in the proof ofTheorem 3.12 in [18]. For an arbitrary subinterval J of [ , ) s we obtain the followingbound (see [18, Inequality (6)]): | S N ( J − λ s ( J ) , ω ) | ≤ ǫ b ( g ) + max I ∈J b , g {| S N ( I − λ s ( I ) , ω ) |} . (5.3)This bound is independent of the choice of J . As a consequence, D N ( ω ) = sup J ∈J | S N ( J − λ s ( J ) , ω ) | ≤ ǫ b ( g ) + D N ; b , g ( ω ) . In the case of the star discrepancy D ∗ N ( ω ) , the intervals J are anchored at the origin.It is easy to see from the proof of Theorem 3.12 in [18] that this fact allows us to replacethe error term ǫ b ( g ) by ǫ ∗ b ( g ) . This finishes the proof. Remark 5.12.
An elementary analytic argument shows that ǫ b ( g ) ≤ sδ g , and ǫ ∗ b ( g ) ≤ sδ g , where δ g = max ≤ i ≤ s b − g i i . Corollary 5.13.
We have the following discretization: D N ( ω ) = sup I ∈J b | S N ( I − λ s ( I ) , ω ) | ,D ∗ N ( ω ) = sup I ∈J ∗ b | S N ( I − λ s ( I ) , ω ) | . We recall that a sequence ω is called uniformly distributed in [ , ) s if and only if ∀ J ∈ J : lim N →∞ S N ( J − λ s ( J ) , ω ) = . (5.4) Corollary 5.14.
It follows from Inequality (5.3) that a sequence ω is uniformly dis-tributed in [ , ) s if and only if ∀ I ∈ J b : lim N →∞ S N ( I − λ s ( I ) , ω ) = . The argument to approximate the discrepancies D N and D ∗ N by a hybrid spectraltest goes as follows. Let b = ( b , . . . , b s ) be a vector of s not necessarily distinctintegers b i ≥
2. As index set, we choose L = N s × N s and observe that L = [ g ∈ N s ( D b ( g ) × ∇ b ( g )) . An index point ( a , d ) ∈ L is called admissible if there exists g ∈ N s such that ( a , d ) ∈ D b ( g ) × ∇ b ( g ) and the interval I a , d ; g belongs to J b , g . The point ( a , d ) ∈ L is called he hybrid spectral test non-admissible otherwise. The reader should note that different admissible points mayproduce the same b -adic interval.As the elements of the functions system F = { ξ ( a , d ) : ( a , d ) ∈ L } , we choose thefunctions I − λ s ( I ) , with I ∈ J b , or the constant function 0, subject to the followingparametrization. For an admissible index ( a , d ) ∈ L , let ξ ( a , d ) = I a , d ; g − λ s ( I a , d ; g ) . It follows that F contains all functions 1 I − λ s ( I ) , with I ∈ J b . If ( a , d ) is non-admissible, define ξ ( a , d ) to be identically 0. Corollary 5.14 implies that F is u.d.d.For a given g ∈ N s , we define the weight function ρ g on L in the following manner.If k ∈ N , with b -adic representation k = k + k b + · · · , we put v b ( k ) = ( , if k = , + max { j : k j = } , if k ≥ . For ( a , d ) ∈ L , a = ( a , . . . , a s ) , d = ( d , . . . , d s ) , let ρ g (( a , d )) = ( , if ( a , d ) ∈ D b ( g ) × ∇ b ( g ) , Q si = b − ( v bi ( a i )+ v bi ( d i )) i , otherwise. (5.5)Further, we choose the maximum norm on R s . Then ρ g is a weight function in thesense of Definition 4.4. Theorem 5.15.
Let L and F be as above. For every ǫ > , there exists an integervector g ∈ N s such that for any sequence ω in [ , ) s , the spectral test σ N ( ω ) of thefirst N elements of ω , with respect to F and ρ g , has the property | σ N ( ω ) − D N ( ω ) | < ǫ. Proof.
Let ǫ > g ∈ N s such thatmax ≤ i ≤ s b − g i i < ǫ/ ( s ) . For the function system F and for the weight function ρ g defined in (5.5), we have σ N ( ω ) = max (cid:8) max {| S N ( ξ ( a , d ) , ω ) | : ( a , d ) ∈ D b ( g ) × ∇ b ( g ) } , sup { ρ g (( a , d )) | S N ( ξ ( a , d ) , ω ) | : ( a , d ) D b ( g ) × ∇ b ( g ) } (cid:9) . We have D N ; b , g ( ω ) = max (cid:8) | S N ( ξ ( a , d ) , ω ) | : ( a , d ) ∈ D b ( g ) × ∇ b ( g ) (cid:9) , and sup (cid:8) ρ g (( a , d )) | S N ( ξ ( a , d ) , ω ) | : ( a , d ) D b ( g ) × ∇ b ( g ) (cid:9) ≤ max ≤ i ≤ s b − − g i i . P. Hellekalek
The choice of g implies D N ; b , g ( ω ) ≤ σ N ( ω ) ≤ D N ; b , g ( ω ) + ǫ/ ( s ) . On the other hand, Theorem 5.11 and Remark 5.12 yield D N ; b , g ( ω ) ≤ D N ( ω ) ≤ D N ; b , g ( ω ) + ǫ/ . The result follows.
Corollary 5.16.
In the case of the star discrepancy, put L = N s and let F be definedaccordingly, such that it contains all the functions I − λ s ( I ) , I ∈ J ∗ b . For every ǫ > , there exists g ∈ N s such that for any sequence ω in [ , ) s , the spectral test σ N ( ω ) of the first N elements of ω , with respect to F and ρ g , has the property | σ N ( ω ) − D ∗ N ( ω ) | < ǫ. Acknowledgments.
My special thanks are due to Roswitha Hofer, University of Linz,for several helpful and most enjoyable discussions.
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