Star discrepancy of generalized two-dimensional Hammersley point sets

Transcription

1 Star discrepancy of generalized two-dimensional Hammersley point sets Henri Faure Astract We generalize to aritrary ases recent results on the star discrepancy of digitally shifted two-dimensional Hammersley point sets in ase y Kritzer, Larcher and Pillichshammer. The key idea is to link our fundamental formula for the discrepancy function of generalized van der Corput sequences to the corresponding quantity for generalized two-dimensional Hammersley point sets. In that way, we can derive precise formulas for the star discrepancy of these point sets and otain simple generalizations which are the est presently known with regard to low star discrepancy. This study is parallel to the recent one y F. Pillichshammer and the author on the L p discrepancy of the same point sets to appear in Monatsh. Math.. AMS suject classification: 11K06, 11K38. Key words: Star discrepancy, Hammersley point sets, van der Corput sequences. 1 Introduction For a point set P = {x 1,..., x N } of N 1 points in the s-dimensional unit-square [0, 1] s and a suinterval J of [0, 1] s, the discrepancy function is defined as EJ; P = AJ; P NV J where AJ; P = #{M; 1 M N, x M J} and V J is the volume of J. In this paper, we are interested in the measures of irregularity of distriution otained with the supremum L norm of the discrepancy function which measures are usually called extreme discrepancies. Classically, one considers the supremum either on all intervals or only on intervals anchored at the origin and defines the extreme discrepancy D and the extreme star discrepancy D of P as DP = sup J EJ; P and D P = sup J EJ ; P where J resp. J is in the shape of s i=1 [y i, z i resp. s i=1 [0, y i. Recall, it is not difficult to prove it, that we always have D P DP s D P. Various monographs have een devoted to the theory of discrepancy in the past decades; we refer to [4] and [19] among the last; see also [0] and [3] for an overview. For an infinite sequence X, we write respectively EJ; N; X = AJ; N; X NV J, D N, X and DN, X for the discrepancy function, the star discrepancy and the discrepancy of its first N points. Also, we do not normalize y the numer of points N and keep the original definition which is etter adapted to our statements. 1

2 In the following, we deal only with dimensions 1 or. In these two dimensions, in order to simplify the writing, we use the notations Ex, y, P := E[0, x [0, y; P and Eα, N, X := E[0, α; N; X for the discrepancy function relative to intervals anchored at the origin. Relation etween sequences and point sets. A general principle also valid in any dimension states the link etween one-dimensional sequences and two-dimentional point sets deduced from them [6, 0]: let X = X N N 1 e an infinite sequence taking its values in [0, 1] and let P e the two-dimentional point set { P = X M, M 1 } ; 1 M N [0, 1]. Then N 1 M N D M, X D P 1 M N D M, X Roth was the first to use such point sets in ase, deduced from the original van der Corput sequence, in his famous paper [1] on the L discrepancy. Then, their utilization with ases and in aritrary dimension has een proposed y Hammersley for numerical integration [14] just efore the introduction y Halton of his well-known sequences [1], which can e seen as multi-dimensional versions of van der Corput sequences. In this paper, we will consider generalized two-dimensional Hammersley point sets deduced from the generalized van der Corput sequences introduced in [5]: Definition 1 generalized van der Corput sequence Let e an integer and let Σ = σ r r 0 e a sequence of permutations of {0, 1,..., 1}. For any integers n and N with n 0 and 1 N n, write N 1 = a r N r in the adic system so that a r N = 0 if r n. Then the generalized van der Corput sequence S Σ associated to Σ is defined y S Σ σ r ar N N :=, for all N 1 r+1 r=0 r=0 in ase If σ r = σ is constant, we write S Σ = S σ. The original van der Corput sequence in ase, S id, is otained with the identical permutation id. Definition generalized Hammersley point set Let e an integer, let S Σ e a generalized van der Corput sequence in ase and let n 0 e an integer. Then the generalized two-dimensional Hammersley point set in ase consisting of n points associated to Σ is defined y { H,n Σ := S Σ N, N 1 } ; 1 N n. n In order to match with the traditional definition of aritrary shifted or not Hammersley point sets which are n-its i.e. whose -adic expansions do not exceed n its, we restrict the infinite sequence of permutations Σ to permutations such that σ r 0 = 0 for all r n, for instance Σ = σ 0,..., σ n 1, id, id, id,.... Hence, the ehavior of H,n Σ will

3 only depend on the finite sequence of n permutations σ = σ 0,..., σ n 1 and we will write H Σ,n =: Hσ,n. Again, if we choose in the aove definition σ j = id for all permutations, then we otain the classical two-dimensional Hammersley point set in ase. In ase, this point set is known as Roth point set since it was first considered y Roth to give a complement to his lower ound on L discrepancy in fact Roth otains the order log N for the star discrepancy of that point set and deduce his complement from it [1, 3 p.74]. A profound result of Schmidt [] improved y Béjian [1] ten years after states that D P > c log N for any two-dimensional point set P of N points with an asolute constant c This lower ound is est possile, apart from the constant c, thanks to Roth and Hammersley point sets. Hence, the goal is to improve the leading constant term in the estimate Olog N. Such improvements are achieved with generalized Hammersley point sets. The main results concerning the star discrepancy of two-dimensional Hammersley point sets are of two kinds: some give exact formulas including complementary terms and the other give formulas for the leading terms within an error not computale, usually lower than a small additive constant. We recall them without going into the details of the formulas. If necessary more information will e given at the right place in the paper. Three papers elong to the first kind: First, Halton and Zarema [13, 1969] otained exact formulas oth for the star discrepancy and the L discrepancy of the Roth point set H,n id and for a modification of this set which, in our terminology, reads as H,n σ with the sequence σ = id, τ, id, τ,..., id, τ where τ is the transposition usually called shift. The leading terms in the formulas for D H,n id and D H,n σ are respectively 1n and 1n. 3 5 Next, L. De Clerck, in a remarkale paper [3, 1986] summing up her thesis in Flemish, has een ale to compute exactly the star discrepancy of the classical Hammersley point set H,n id in an aritrary ase, giving at the same time the extremal intervals for which the discrepancy is reached. The leading term in the formula for D H,n id is respectively 1 n if is odd and n if is even Finally, Larcher and Pillichshammer, as a y-product of their study of 0, m, nets in ase [18, 003], gave a new proof of the exact formula found y Halton and Zarema and De Clerck for D H,n. id There is an interesting discussion in this paper concerning the discrete discrepancy instead of reals, the supremum is taken over n-its numers: the authors oserve that it differs from D at most y the almost negligile quantity and that it seems for nets to e the more natural measure for the irregularities of distritution. Moreover, their proofs of the exact formulas for these two discrepancies of H,n id clearly illustrate that the discrete discrepancy is much easier to handle than D, in spite of the minimal difference etween them [18, Section 4 p.399]. The other papers elong to the second kind: First, the author [6, 1986], in parallel with the work of De Clerck, deduced approximate formulas from his study of generalized van der Corput sequences in aritrary ases: general upper ounds valid for generalized Hammersley point sets H,n Σ not necessarily n- its nor limited to n points Theorems 1 and and more specific estimates for D H,n id giving the results of De clerck within an additive term c n [0, ] Theorem 3. As a consequence of Theorem, we got the smallest star discrepancy for finite two-dimensional 3

4 point sets with the leading constant in ase 1 compare with the lower ound 0.06, ut with a complementary term in O log N. Next, Kritzer [15, 006], extending the fundamental Theorem 1 of [18], proved that H,n id is the worst distriuted net among shifted digital 0, m, nets in ase and also that digitally shifted Hammersley nets in ase are still etter than H,n. id Among them, with the net H,n, iτ where iτ = id,..., id, τ,..., τ half and half aout, he otained D H,n iτ n + c with c < 4, which gives the leading constant Next, Kritzer, Larcher and Pillichshammer [16, 007], continuing the preceding paper, made a very detailed investigation of sums of distances to the nearest integer and deduced an explicit formula Theorem 3 for the star discrepancy of shifted Hammersley point sets in ase, within an additive term c n [0, ] in fact, they computed exactly the discrete discrepancy of these sets, see aove. Using that formula, they were ale to generalize the result of Kritzer with the sequence σ = id,..., id, τ,..., τ,..., id,..., id, τ,..., τ the same numer of permutations in each lock. This formula also allows them to recover two results they proved efore in the paper with less accuracy Theorems 1 and. We will generalize these results to aritrary ases in Section 4 of the present paper. Finally, at the end of our paper [10, 008], we derive improved results for generalized Hammersley point sets from improvements on van der Corput sequences, in the same way as in [6]. For instance, we show that the so-called linear digit scramlings give very good leading terms around 0.3 in any given prime ase. But the est constants in ases, 3 and 1 are still otained with a complementary term in O log N due to the definition of the related van der Corput sequences [10, Section 5.]. It was one of the aims of the present paper to remove this restriction, as announced in the last lines of [10]. The paper is organized as follows: Section introduces the necessary material for the proofs of theorems of Sections 3 to 5. In Section 3, we prove a general formula for generalized two-dimensional Hammersley point sets in aritrary ase which is the analog of [16, Lemma 1] in ase. Section 4 is devoted to results otained with the identical permutation: generalizations to ase of [15, Theorem 3.] and [16, Theorems 1 and ]. Finally, in Section 5 we deal with aritrary permutations in ase and otain simple generalizations of two-dimensional Hammersley point sets which are the est presently known with regard to low star discrepancy, hence carrying out the proposal at the end of [10]. Prerequisites In this section, we recall the main results we will need for the proofs of our theorems. They concern generalized van der Corput sequences and come from [5, ], in French. Another reference is [9] in English in which, even if the suject is different, many definitions are the same and formulas are quite similar see also [8]. 4

5 .1 Functions ϕ σ,h related to a pair, σ σ0 Let σ e a permutation of {0, 1,..., 1} and let Z σ σ 1 =,, h {0, 1,..., 1} and x [k 1/, k/ where k {1,..., } we define [ A 0, h ; k; Z ϕ σ,hx = σ hx if 0 h σk 1, hx A[ h, 1 ; k; Z σ if σk 1 < h <.. For Further, the function ϕ σ,h is extended to R y periodicity. For future use in Lemma, ϕ σ,h denotes its right derivative. Note that ϕ σ,0 = 0 for any σ and that ϕσ,h 0 = 0 for any σ and any h. As a matter of fact, the functions ϕ σ,h are linearizations of remainders related to Zσ. They are the fundamental tool for the study of irregularities of distriution of sequences S Σ. Actually, they give rise to other functions, depending only on, σ, according to the notion of discrepancy we are dealing with. Presently, we will need ψ σ,+ = 0 h 1 ϕσ,h, ψ σ, = 0 h 1 ϕσ,h and ψ σ = ψ σ,+ + ψ σ, = 0 h<h 1 ϕσ,h ϕσ,h. Many properties of ψ functions are given in [5, 3.]. Recall for future use that they are continuous and convex on intervals [ k 1, ] k 1 k. An important permutation which plays a leading part in constructions improving discrepancy [5, 6, 10, 11] is the so-called swapping permutation τ defined y τk = k 1 0 k 1. For a given permutation σ, τ swaps ψ σ,+ and ψ σ, which means that ψ τ σ,+ = ψ σ, and ψ τ σ, = ψ σ,+ see [5, 4.4.1] and also [11, Lemma 1] when σ = id. We will use it in Sections 4 and 5. In the special case =, we only have two permutations which give either ϕ σ,1 = if σ = id or ϕ σ,1 = if σ = τ, where is the distance to the nearest integer. Naturally, this function takes a central place in the studies in ase of Kritzer, Larcher and Pillichshammer, especially [18, 15, 16] for the star discrepancy. Note that in ase, the swap τ is usually called the shift ecause τk = k + 1 in F.. Exact formulas for the discrepancies of S Σ For any infinite sequence X, we define the positive and negative discrepancies as D + N, X := sup Eα, N, X and D N, X := sup Eα, N, X. 0 α 1 0 α 1 Then [5, Section 3.3.6], for all integers n 1 and N with 1 N n, we have N D + N, S Σ = + N N n ψ σ j 1,+ j j=n+1 σ j 1 0 j and D N, S Σ = ψ σ j 1, N + N j j=n+1 σ j 1 0 j. 3 5

6 Recall that D = D +, D and D = D + + D, so that we also have exact formulas for D N, S Σ and DN, SΣ. In our paper [5], formulas and 3 needed a lot of lemmas to e proved ecause, at this time, we did not have an exact formula for the discrepancy function Eα, N, S Σ. Indeed, they can e otained more simply with the help of such a formula see Section.4 elow which also will play a leading part in the study of generalized Hammersley point sets..3 Estimates for sums of ψ functions In order to compare generalized Hammersley point sets and find the est possile with respect to low discrepancy, we will need estimations of sums of ψ functions which appear in and 3. We collect the necessary information in the following lemma, in which the notation ψ stands either for ψ σ or ψσ,+ or ψ σ,. Lemma 1 i For any integer n 1, let d n := x R x d n ψ and α := inf j n 1 n. Then d n α = lim n n and there exists β n [0, 1] such that d n = αn + β n for all n 1. n 1 ii For any x [0, 1] and any n 1, let F n x := ψx k. Then d n = k=0 and for all integers ν, a with 1 a ν, we have 1 a ν F ν α. ν 1 x [0,1] F n x Proof. The proof is scattered in [5, 4.. and 5..1] see also [, ]. Practically, the exact computation of α is possile in some specific cases, for instance for small or for the identical permutation. In that cases, the notion of dominating interval is very useful to reduce the prolem and to formulate an induction hypothesis for the value of d n see [5, 5.3] and also [8, 4.4]. But in any case, it is easy to otain upper and lower ounds d with small values of n and ν in the formulas α = inf nn n 1 and 1F a ν ν 1 α aove. ν.4 Descent Lemma This is the fundamental Lemma 5. of [] and, in the adaptation to 0, 1 sequences, Lemma 6. of [9]. The foremost version was found for the precise study of the L discrepancy of the van der Corput sequence and its symmetrisized version in ase two [7]. Lemma Let n 1, N and λ e integers with 1 N n and 1 λ < n and let λ = λ 1 n λ n 1 + λ n e the -adic expansion of λ. Then, the discrepancy function of the first N points of S Σ on [0, λ [ satisfies n λ E, N, SΣ n = ϕ σ j 1 N,ε j, 4 j the ε j s eing defined step y step as follows: ε n := η n := λ n and, for 1 j < n, η j := λ j + η j ϕ σ j 1 N,ε j+1, 5 j+1 ε j := η j if 0 η j < and ε j := 0 if η j =. 6

7 Remark 1 For λ = n the formula is trivially true with ε j = 0 for all 1 j n since E 1, N, S Σ n = N N = 0 and for N = n also since again oth terms are naught E λ, 1, S Σ n = λ λ = 0 and the functions ϕ are zero on integers. Proof. We refer to [] for the proof. The analog for 0, 1 sequences in [9], in English, is more complicated ut follows the same framework: step y step, we otain the discrepancy function E λ, N, S Σ n y means of discrepancy functions with more and more rough intervals and with less and less points; at each step, the difference etween the discrepancy functions is under control with the help of the functions ϕ σ,h while the relation etween the intervals depends on the right derivatives of these functions. At the end of the descent, we otain Equation 4. 3 A general formula for D H σ,n Now, we deal with generalized Hammersley points sets H σ,n and first link the discrepancy function of H,n σ on the two-dimensional interval [0, λ [0, N to the discrepancy n n function of the first N points of S Σ on [0, λ recall from the definition of n Hσ,n that Σ = σ 0,..., σ n 1, id, id,... and σ = σ 0,..., σ n 1. Lemma 3 Let n 1, N and λ e integers with 1 N n and 1 λ < n. Then, λ E, N λ n, n Hσ,n = E, N, n SΣ Proof. The proof is almost immediate: from the definition of H,n σ the counting functions A λ, N, H σ n n,n and A λ, N, S Σ n take the same value and on the other hand, n λ N n = n N λ, so that the discrepancy functions are equal. n Note that Lemma 3 together with Lemma is very close to [11, Lemma 1] only the form of digits ε j in Lemma differs. Next, we prove an exact formula for the so-called star discrete discrepancy see [18, Section 4]: Lemma 4 Let n 1 e an integer. Then, the star discrete discrepancy of H σ,n, λ E, N n,n, n Hσ = 1 λ,n n 1 N n ψ σ j 1,+ N j, 1 N n ψ σ j 1, N j. 6 Proof. From Lemmas and 3 we first have the ε j eing defined y the formulas 5: E λ, N n, n Hσ,n = ϕ σ j 1,ε j N j for all 1 λ, N n. 7

8 Now, from the definitions of ψ σ,+ E λ, N n, n Hσ,n and ψ σ,, we deduce that ψ σ j 1,+ N and E j λ, N n, n Hσ,n ψ σ j 1, N. j But these upper ounds are reached for some λ s y using the reverse algorithm of the construction 5 of the ε j from the λ j as we did in the proof of Theorem 1 of [9, 6.4]. We recall the proof for the sake of completeness: for 1 j n, let p j 0 p j < e an integer such that ψ σ j 1,+ N = ϕ σ j 1 j,p j N, so that we have a fixed sequence ε j j = p j from which we can deduce an integer λ = λ 1 n λ n satisfying Eλ n, N n, H,n σ = n ϕσ j 1,p j N j = n ψσ j 1,+ N j. Indeed, using the formulas of 5 for η j and ε j, we uild λ step y step, first y setting λ n := η n := p n and then, if η j and λ j are achieved, y using the reverse algorithm: if η j + 1 ϕσ j 1,p j N j = 0 then λ j 1 := η j 1 := p j 1 else i.e. η j + 1 ϕσ j 1,p j N j = 1 then if p j 1 1 then η j 1 := p j 1 and λ j 1 := η j 1 1 else i.e. p j 1 = 0 then η j 1 := and λ j 1 := η j 1 1. The same is valid for ψ σ, λ E 1 λ n, N n, n Hσ,n λ E 1 λ n, N n, n Hσ,n λ 1 λ E n, N n, n Hσ,n 1 λ,n n and therefore we have proved that = = ψ σ j 1,+ N =: D + j n N, S Σ and ψ σ j 1, N =: D j n N, S Σ, which implies = D n + N, S Σ, Dn N, S Σ, so that λ E, N n,,n n Hσ = D + n N, S Σ 1 N n, Dn N, S Σ. D n + N, S Σ and D n N, S Σ are called the discrete positive and negative discrepancies of S Σ. Finally, discussing the two possiilities for the integer N 0 which achieves the value of 1 N n D + n N, S Σ, D n N, S Σ, we see that D + n N, S Σ 1 N n, Dn N, S Σ = D + n N, S Σ 1 N n, D n N, S Σ 1 N n and Lemma 4 follows. Theorem 1 For any integer n 1 and any σ = σ 0,..., σ n 1 we have, with some c n [0, ], D H,n σ = ψ σ j 1,+ N 1 N n, ψ σ j 1, N 1 N n + c n. j j 8

9 Proof. For x [0, 1], let xn := xn, so that xn 1 x < xn. Since all points of n n H,n σ have n-it coordinates i.e. are of the form α/n, we have Ex, y, H σ,n = Exn, yn, H σ,n + n xnyn xy. From the definition of xn we deduce that 0 < n xnyn xy xn + yn 1 < n and therefore there exists c n [0, ] such that D H σ,n = sup Ex, y, H,n σ = x,y [0,1] 1 λ,n n λ E, N n,n, n Hσ + c n. By Lemma 4, the proof of Theorem 1 is complete. Corollary 1 For any integer n 1 and any σ = σ 0,..., σ n 1 we have, with some c n [0, ], D H,n σ D H,n id + c n. Proof. This corollary results from the following properties of ψ functions: ψ σ,+ ψ id, ψ σ, ψ id, ψid,+ = ψ id and ψ id, = 0. The complementary term c n could e removed with more effort in the context of 0, m, nets. 4 Swapping with the identical permutation In this section, we consider generalized Hammersley point sets H,n σ where σ = σ 0,..., σ n 1 {id, τ} n, i.e. where the permutations σ j are either id or τ id = τ, τ eing the swapping permutation defined y τk = k 1 0 k 1. The more general case with an aritrary permutation σ instead of id will e handled in Section 5, ut recall see Section.1 that the interest of permutation τ is to swap ψ σ,+ ψ σ, ψ τ,+ for ψ σ,+ = ψ id, i.e. to give ψ τ σ,+ = 0 and ψ τ, = ψ id,+ = ψ σ, = ψ id. and ψ τ σ, = ψ σ,+ for ψ σ, and which, when σ = id, reads as {}}{{}}{ First let us consider the sequence iτ = id,..., id, τ,... τ if n is even and iτ = n 1 n+1 {}}{{}}{ id,..., id, τ,... τ, if n is odd, like Kritzer did in ase [15]. Applying Theorem 1, we can easily extend his result [15, Theorem 3. and proposition 3.1] to aritrary ases: Theorem For any integer n 1 we have, with some c n [0, 3], if is odd: if is even: D H iτ,n = D H iτ,n = n 1 8 n + c n if n is even 1 8 n c n if n is odd n + c n if n is even n c n if n is odd. 9 n

10 Proof. We first deal with an even integer, say n = m, and apply Theorem 1 with ψ σ j 1,+ = ψ id,+ = ψ id, ψσ j 1, = ψ id, = 0 if 1 j m and ψ σ j 1,+ = ψ τ,+ = 0, ψ σ j 1, = ψ τ, = ψ id if m + 1 j m. The two sums of n terms reduce to sums of m terms and we get with some c n [0, ] D H,n iτ = 1 N n ψ id N, j 1 N n j=m+1 ψ id N + c n. We see that the alance etween the two sums of Theorem 1 is perfect, which will divide y the discrepancy of the usual Hammersley net. Indeed, we know the ehavior of sums of ψ functions, see Section.3, and since these functions are continuous, 1-periodic and convex on intervals [ k 1, ] k 1 k, the imum for x R in Lemma 1 i is reduced to the imum for integers N [1, n ] only. Hence, applying Lemma 1 i, we otain note that the two first equalities elow hold as well from the same property: 1 N n ψ id N = j 1 N n j=m+1 ψ id N j = 1 N m with β m [0, 1]. We have computed the constants α id α id = 1 4 if is odd and α id = ψ id j N j = α id m + β m in [5, 5.5] and we found that if is even. This achieves the proof when n is even with c n = c n + β m. In the case where n = m + 1 is odd, in the same way we otain D H,n iτ = 1 N n ψ id N, j m+1 1 N n j=m+1 ψ id N + c n. Here, the alance etween the two sums of Theorem 1 differs y one term only, which explains the factor n + 1 instead of n in the result for odd n. Corollary Asymptotically, we get j D H,n iτ lim = 1 n log n 8 log if is odd and D H,n iτ lim = n log n log if is even. Remark The interval for c n could e reduced, in particular with the exact values for β m. Of course, we recover the result of Kritzer [15] and Kritzer Larcher Pillichshammer [16], with the same sequence iτ, in the case of =. The est constant is otained for 1 = 3 with 4 log 3 = , whereas for = we only have 1 = log Now, we will show that the choice of the sequence iτ in the set {id, τ} n of sequences σ = σ 0,..., σ n 1 is est possile in the sense that the leading terms in Theorem cannot e made smaller whatever the σ j 1 {id, τ}, 1 j n, can e. 10

11 Theorem 3 For any integer n 1 and any σ {id, τ} n we have D H,n σ lim 1 n log n 8 log if is odd and D H,n σ lim n log n log if is even. Proof. We use the properties of infinite sequences S Σ together with the first inequality in relation 1 to derive the desired lower ound for D H,n σ. Recall from Definition that Σ = σ 0,..., σ n 1, id, id... with σ j 1 {id, τ}. This implies that for all j, 1 j n, we have ψ σ j 1 = ψ id = ψid, so that, from formulas of Section., we deduce DN, S Σ = DN, Sid. Recall too that in one dimension we have D D. Now, we can give the proof: D H σ,n 1 N D N, S Σ n 1 DN, Sid 1 N n 1 αid n and the result follows from the reminders in the proof of Theorem. In ase, Theorem 3 has een shown in [16] y other arguments involving more computations. Another question raised and solved in ase in [16] is the following: Does the star discrepancy D H,n σ is independent of the distriution of id and τ in the sequence σ = σ 0,..., σ n 1 {id, τ} n and depends only on the numer of id and τ as it is the case with the L discrepancy see [17] and [11]? In aritrary ase, the answer is no like in ase, with the same counter-example as in [16]: the sequence id, τ, id, τ,..., id, τ. To prove this result see Theorem 4 elow we will utilize the notion of intrication of permutations introduced in [5, Section 3.4.3]: Definition 3 The intrication of two pairs, σ and c, ρ is the pair c, σ.ρ defined y σ.ρl = cσh + ρk with l = k + h, 0 h < and 0 k < c euclidean division of l y, 0 l < c. Lemma 5 i The function ψ σ.ρ,+ c satifies the relation ψ σ.ρ,+ c x = ψ σ,+ cx + ψc ρ,+ x for all x R and the same relation is also valid for ψ σ.ρ, c and ψ σ.ρ c. ii Let Σ = σ j j 0 and T = τ j j 0 e two aritrary sequences of permutations of {0, 1,..., 1}. If S P is the sequence defined y P = ρ j j 0 with ρ j = σ j if j is even and ρ j = τ j if j is odd, then S P = SΣ.T where Σ.T := σ j.τ j j 0. Proof. Part i of this lemma is part i of [5, Proposition 3.4.3], proved y starting from the definition of ψ σ.ρ,+ c and enumerating the different cases. The proof of ii follows the same lines as [5, Proposition ii], ut with a fixed ase instead of different ases and with two aritrary sequences of permutations instead of two constant sequences of permutations: starting from the definition of S Σ.T, it suffices to write the relations etween the expansions of N 1 in ases and to recover S P with Definition 3. 11

12 Theorem 4 For any even integer n, let ĩτ = id, τ, id, τ,..., id, τ {id, τ}n. Then, with some c n [0, 3], we have 1 + n + c n if is odd, D iτ H f 8 + 1,n = n + c n if is even. These constants are greater than 1 8 aove is no. and, hence the answer to the question Proof. Let n = m. First, applying Lemma 5 ii with σ j = id for all j 0, τ j = τ if 0 j m 1 and τ j = id if j m, so that P = id, τ, id, τ,..., id, τ, id, id,..., we get S P = S Σ.T with Σ.T = id.τ, id.τ,..., id.τ, id.id, id.id,... m times id.τ and then id.id only. Therefore, from the definition of generalized Hammersley point sets, we have { iτ H f,n = S P N, N 1 } { ; 1 N n = S Σ.T n N, N 1 } ; 1 N m, m so that H f iτ,n = H i.τ,m Next, we apply Theorem 1 to H i.τ D H i.τ,m = 1 N m with i.τ := id.τ, id.τ,..., id.τ. }{{} m,m and get, with some c m [0, ], ψ id.τ,+ N j, 1 N m ψ id.τ, N j + c m. But from Lemma 5 i, we deduce that ψ id.τ,+ x = ψ id x and ψid.τ, x = ψ id x for N all x R. Hence it remains to compute the two ima: and 1 N m N ψ id. j 1 N m Finally, thanks to the periodicity of ψ functions, in fact, the two sums in these ima have the same ehavior and we will study the second one only. For the same reasons as in the proof of Theorem, we have see Lemma 1 for notations 1 N m N ψ id = j x R ψ id j x ψ id = F mx = d j m = αm + β m. x [0,1] The calculation of α and β m is parallel to that for sequences S id performed in [5, 5.5] and since it comes under specific computations according to the parity of, we defer it to the following Proposition 1 which will complete the proof of Theorem 4. Remark 3 Before starting with Proposition 1 and its proof, we want to precise two things: first note that we deal with the function ψ := ψ id in ase, i.e. the 1-periodic function which is equal to the restriction of ψ id on intervals [ k 1, k ] 1 k. Secondly, we oserve that it should e easy otain the leading constants as lower ounds 1

13 1 for α with the help of Lemma 1 ii: if is odd, taking ν = 1 and a = 1 + lead to α and if is even, taking ν = and a = would would lead to α we omit the proof since we will get the equality elow. Therefore, the answer to the question aove would e no, without anymore computations. But, like Kritzer, Larcher and Pillichshammer did in the case of ase see [16, Section 4], we think it is interesting in itself to otain the exact value of α together with the remaining term β m. Proposition 1 For any integer m 1 we have 1 + N m ψ id N m = m j m m m if is odd, if is even. Proof. According to Lemma 1, we have to compute d m = αm + β m = F m x with F m x = m 1 k=0 x [0,1] ψ x k. We proceed y induction on m 1. We have found the induction hypothesis y plotting the first graphs of the functions ψ = ψ id for small ases 7 and looking at the dominating intervals, i.e. intervals I k0 = [ k 0 1, k 0 ] such that ψ is dominated y its restriction to I k0 within a translation or a symmetry. Then verifying the induction hypothesis reduces to add ψ x m to F m x and check the formulas for m + 1. In the following, we give in detail the induction hypothesis ut we leave out the tedious verifications which are long ut straightforward. For the sake of completeness, recall the formulas for ψ id [5, 5.5.1]: ψ id x = k1 x, k 1x for all x [k/, k + 1/] and integers k [0, 1]. The case of an odd ase. There is one dominating half interval [u m, u m + 1 ] with u m m = m On this interval, F m is the linear function F m x = q m x + F m u m with q m = m 1 + 1, 1 + F m u m = m and d m m = F mx = F m u m. x [0,1] The case of an even ase. There is one dominating half interval [u m, u m + 1m m ] with u m = 1m m On this interval, F m x = q m x + F m u m with q m = 1 m m + 1 m 1, 3 F m u m = m m and d m m = F mx = F m u m. x [0,1] These formulas achieve the proof of Proposition 1 and therefore the proof of Theorem 4 too. 13

14 Corollary 3 Asymptotically, we get with even n 1 + D iτ H f,n lim = n log n log log if is odd and if is even. Remark 4 Of course, for = we recover the result of [16, end of Section 4] with the constant 5 1 In fact, this result was known for a long time with much more precision since Halton and Zarema [13, 1969] otained exact formulas for D H,n id and D iτ H f,n after a lot of technical computations see [18, Sections 1 and 4] for comments on this paper. 5 Swapping with an aritrary permutation In this section, we fix an aritrary permutation σ of {0, 1,..., 1} and consider sequences produced y swapping σ with τ, that is we mix the permutations σ and σ := τ σ to otain sequences σ = σ 0,..., σ n 1 {σ, σ} n. The situation is not so clear as with {}}{{}}{ the identity and we will only consider sequences σσ = σ,..., σ, σ,... σ if n is even and n 1 n+1 {}}{{}}{ σσ = σ,..., σ, σ,... σ if n is odd. This choice permits to improve upon the discrepancy, ut until now we are not ale to prove it is the est, like in Section 4 with the identity and Theorem 3. Let us recall two notations adapted to the present context see Lemma 1 efore to state the result for the sequence σσ. Set α σ,+ := inf 1 n 1 x R n x ψ σ,+ and α σ, j := inf 1 n 1 x R n Theorem 5. For any integer n 1 we have, with some c n [0, 4], D H,n σσ = D H,n σσ = α σ,+ α σ,+ 1 N n n + α σ, n + c n if n is even + α σ, n c n if n is odd. ψ σ,+ n x ψ σ,. Then : j Proof. We condider only the case of an even integer n = m the case of an odd n is handled in the same way as in Theorem. Applying Theorem 1 with ψ σ,+ = ψ τ σ,+ = ψ σ, and ψ σ, = ψ τ σ, = ψ σ,+, we otain, with some c n [0, ], m N N + 1 N n m ψ σ, j N + j j=m+1 j=m+1 ψ σ, ψ σ,+ j + c n. N j 14

15 Now, of course we have with ovious A N and B N A 1 N n N + B N A 1 N n N + B N. 1 N n But, due to the properties of the ψ functions see [5, Lemma 4..1], we also have A 1 N n N + B 1 N n N A 1 N n N + B N + 1. Discussing the ima and the sums like in the proof of Theorem and applying Lemma 1 with functions ψ σ,+ and ψ σ,, we otain with β m + and βm [0, 1] α σ,+ m + β m + + α σ, m + βm 1 A 1 N n N + B N α σ,+ m + β m + + α σ, m + βm. Finally, ringing ack these ounds in D H,n σσ, we get the desired result with c n = c n + β m + + βm [0, 4]. Corollary 4 Asymptotically, we get D H,n σσ lim n log n = ασ,+ + α σ, log Remark 5 Of course, we recover Corollary when σ = id. Until now, the est result coming from [5, Théorème 5] is otained for = 1 and the permutation σ = product of cycles with D H1,n σσ lim n log 1 n = ασ,+ 1 + α σ, 1 log 1 = log 1 =.3..., 1 a it etter than = 3 and σ = id with = Even if such improvements 4 log 3 seem small, they concern the leading constants in discrepancy formulas and we think it is more important to improve upon these constants rather than to search for exact formulas or to reduce the complementary terms c n in estimations. References [1] R. Béjian: Minoration de la discrépance d une suite quelconque sur T. Acta Arith. 41: 185 0, 198. [] H. Chaix and H. Faure: Discrépance et diaphonie en dimension un. Acta Arith. 63: , [3] L. De Clerck: A method for the exact calculation of the star-discrepancy of plane sets applied to the sequences of Hammersley. Monatsh. Math. 101: 61 78, [4] M. Drmota, and R.F. Tichy, Sequences, Discrepancies and Applications. Lecture Notes in Mathematics 1651, Springer-Verlag, Berlin, [5] H. Faure: Discrépance de suites associées à un système de numération en dimension un. Bull. Soc. Math. France 109: ,

16 [6] H. Faure: On the star-discrepancy of generalized Hammersley sequences in two dimensions. Monatsh. Math. 101: , [7] H. Faure: Discrépance quadratique de la suite de van der Corput et de sa symétrique. Acta Arith. 55: , [8] H. Faure: Good permutations for extreme discrepancy. J. Numer Theory. 41: 47 56, 199. [9] H. Faure: Discrepancy and diaphony of digital 0, 1-sequences in prime ases. Acta Arith. 117: , 005. [10] H. Faure: Improvements on low discrepancy one-dimensional sequences and twodimensional point sets. In: Keller, A., Heinrich, S. and Niederreiter, H. Eds. Monte Carlo and Quasi-Monte Carlo Methods 006. Springer, , 008. [11] H. Faure and F. Pillichshammer: L p discrepancy of generalized two-dimensional Hammersley point sets. Monatsh. Math., to appear. [1] J. H. Halton: On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numerische Mathematik : , [13] J.H. Halton, and S.K. Zarema: The extreme and the L discrepancies of some plane sets. Monatsh. Math. 73: , [14] J. M. Hammersley: Monte Carlo methods for solving multivariale prolems. Ann. New York Acad. Sci. 86: , [15] P. Kritzer: On some remarkale properties of the two-dimensional Hammersley point set in ase. J. Théor. Nomres Bordeaux 18: 03 1, 006. [16] P. Kritzer, G. Larcher and F. Pillichshammer: A thorough analysis of the discrepancy of shifted Hammersley and van der Corput point sets. Ann. Mat. Pura Appl. 186: 9 50, 007. [17] P. Kritzer and F. Pillichshammer: An exact formula for the L discrepancy of the shifted Hammersley point set. Uniform Distriution Theory 1: 1 13, 006. [18] G. Larcher and F. Pillichshammer: Sums of distances to the nearest integer and the discrepancy of digital nets. Acta Arith : , 003. [19] J. Matoušek: Geometric Discrepancy. Algorithms and Cominatorics 18. Springer, Berlin, [0] H. Niederreiter: Random Numer Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 199. [1] K.F. Roth: On irregularities of distriution. Mathematika 1: 73 79, [] W.M. Schmidt: Irregularities of distriution VII. Acta Arith. 1: 45 50,

17 [3] O. Strauch and Š. Poruský, Distriution of Sequences: A Sampler. Peter Lang, Bern, 005. Henri Faure, Institut de Mathématiques de Luminy, U.M.R. 606 CNRS, 163 avenue de Luminy, case 907, 1388 Marseille Cedex 09, France et Université Paul Cézanne. faure@iml.univ-mrs.fr 17