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....to evaluate the error term in the uniform distribution of 0 # 1: we have = I O(N # ) and this estimate is sharp (errors of size comparable to N # actually occur, when 0 1. For more on distribution of sequences modulo 1, see, for example, We] HW, Ch. XXIII] Sa] and [KN]. 2 Ergodic theory In this section we prove a general form of Theorem 1.4 on the uniform distribution of horocycle sections # : S E. This ergodic theoretic result will allow us to relate the gap distribution for # n to random lattices, as sketched in the Introduction. 2.1 The a#ne group of ....

L. Kuipers and H. Niederreiter. Uniform distribution of sequences. Wiley, 1974. 45

....sequence) if it is a b Benford sequence for all b 1 . The most basic tool in this paper is the following direct correspondence between Benford sequences and uniform distribution modulo one, which allows application of the powerful classical tools for uniform distribution of sequences (e.g. [DT, KN]) Proposition 2.2 ( D] A sequence (x n ) n#N0 of real numbers is a b Benford sequence if and only if (log b n#N0 is uniformly distributed modulo one. Henceforth, the term uniformly distributed modulo one will be abbreviated as u.d. mod 1. An immediate consequence of Proposition 2.2 is ....

....(S j ) On the other hand, for x su#ciently close to 0, the orbit O T (x) is a b Benford sequence precisely if OS Example 5.4. Corollary 5.2 reduces the question of whether (5.1) generates Benford sequences to a problem of uniform distribution. Using standard techniques from that theory (e.g. [KN]) it is straightforward to prove that for the classes of sequences (# j ) listed below, O T (x) is a strict Benford sequence for all initial points x close to infinity or 0, respectively. The proof of (i) uses [KN, Thm. 3.3] ii) uses [KN, Thm. 2.7] and Euler s summation formula; iii) is ....

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Kuipers, L. and Niederreiter, H. (1974) Uniform Distribution of Sequences. Wiley, New York.

....be simultaneously stratified with respect to every hyper rectangular subset of [0, 1) yet it is interesting to ask how far we might be able to go in that direction. This is a problem that has been studied since Weyl [44] originated his theory of uniform distribution. Kuipers and Niederreiter [21] summarize that theory. Let a and c be points in [0, 1) for which a c holds componentwise, and then let [a, c) denote the box of points x where a x c holds componentwise. We use c) to denote the d dimensional volume of this box. An infinite sequence of points x 1 , x 2 , # is ....

L. Kuipers and H. Niederreiter. Uniform Distribution of Sequences. John Wiley and Son, New York, 1976.

....of the Cspace topology on the sampling criteria. One of our motivations for choosing the lattices of [44] was their design for rapid integration of periodic functions (the integration occurs over a torus) Although the general theory encompasses arbitrary measures in topological spaces [36], well known sample sets such as the Hammersley Halton sequences are designed for low discrepancy over [0; 1] There are exceptions, such as the low dispersion sequence given for a d dimensional torus in [12] page 115) If the boundary identi cations are taken into account, it should be ....

H. Niederreiter and L. Kuipers. Uniform Distribution of Sequences. John Wiley and Sons, New York, 1974.

..... is irrational, then E n = 1 2 as n 1. Sketch of Proof. Here we only sketch how to estimate the main part of R n delaying the derivation of E n the next section. In the derivation of E n , we shall use discrepancy theory and uniformly distributed sequences modulo 1 (cf. [3, 8, 15]) Our proof rst approximates the binomial distribution by its Gauss density, and then estimates the sum by the Gaussian integral, coupling with large deviations of the binomial distribution. By Stirling s formula, we have 1 2 log n 1 2 O( jxj jxj for k = n x (1 )n and x ....

.... p n (k) p n (k; P e (a; b) a 2 ) b 2 ) a; b) The main result, formulated below, is a consequence of applying analytic tools such as theory of distribution of sequences modulo 1 and Fourier series, as already advocated in [14] The interested reader is referred to [3, 8, 15]. Theorem 2 (i) If = log for some positive integers M;N with gcd(M; N) 1, then (4) holds, that is, E n = 1 2 1 2 as n 1, where GM (y) 5 . is irrational then E n = 1 2 as n 1. We start with the following lemma. Lemma 1 Set : log (1 ) log(1 ....

L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences . John Wiley & Sons, New York 1974.

....the following Theorem gives a practical way to estimate the value of the discrepancy of a sequence. Theorem 3.1 (Erdos Turan) For any finite sequence (x m ) 1#m#N of real numbers and any positive integer n, we have: # # # 2#ihxm # # # # . This result is proved in [15] for example. Now, the problem is shifted to an asymptotic analysis of the exponential sum. Fortunately, this can be done for a sequence (x m ) m#N # defined by a polynomial: Definition 3.2 Let d, N be positive integral numbers. Define the quantity # by: # = d 1 . For each value of N , divide ....

....the following notations: N # , # m = 2# 2# = 2# PH0 ,T (m) 5 where PH 0 ,T is a polynomial defined by: PH 0 ,T (x) p k x R. The polynomial PH 0 ,T will play the role of the polynomial A. As the coe#cient p r is irrational, this sequence is uniformly distributed mod(1) e.g. [15]) The two following steps aim at giving an estimate of the discrepancy of the sequence (P H 0 ,T (m) Lemma 3.2 Let (a 1 , a d ) be a d uple of R with rational coe#cients. There exists an integer N 0 such that: N 0 , a 1 , a d ) 1 (N) Suppose now, there exists an r ....

Kuipers & Niederreiter, Uniform distribution of sequences, Wiley Interscience (1974).

....by transseries. In this article, we make a first step towards the treatment of functions involving oscillatory behaviour. The structure of this paper is as follows: in section 2, we recall a classical density theorem for linear curves on the n dimensional torus (see for example [Kok 37] or [KN 74] In section 3, this theorem is generalized to more general classes of curves on the torus. In section 4, we study exp log functions at infinity: an exp log function is a function which is built up from the rationals Q and x, using the field operations, exponentiation and logarithm. An exp log ....

L. Kuipers, H. Niederreiter. Uniform distribution sequences. Wiley (New York).

.... 1 Gm 1 jH n j Gm 2 Gm 1 Gm 1 Gm : Remark 2. The proof of Proposition 1 works for any subsequence (H (n) n of (H n ) n such that ( n) log j j) is dense modulo 1. In particular, it holds for (n) P (n) deg P 1; P 2 Z[x] See Kuipers and Niederreiter [Ku N]) 3. Summatory formul Let us recall some recent results of Cateland and Tenenbaum. For 2 and = 0 ; 1 ) belonging to L (q) 0; q 1 ] n = P k k (n)q k 2 N , Cateland de nes k (n) k (n) k 1 (n) k (n; 1 if k (n) 0 ....

L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, John Wiley, New York (1974).

....if the sequence (a n ) is uniformly distributed modulo 1, then almost uniformly distributed modulo 1. On the contrary, if n j 1 n j = o(n j ) then almost uniformly distributed modulo 1 implies uniformly distributed modulo 1. Using the classical method of uniform distribution theory (see e.g. [6]) we can show the following Proposition 1. The sequence (a n ) n = 1; 2; is almost uniformly distributed modulo 1 if and only if there exist a strictly increasing sequence of natural numbers (n j ) j = 1; 2; such that for every real valued continuous function on the interval [0; ....

L. Kuipers and H. Niederreiter, Uniform distribution of sequences. , Pure and Applied Math., John Wiley & Sons, (1974)

....at random from Z # q . To show the insecurity of DSA with partially known nonces, Nguyen and Shparlinski [23] generalized the previous result to cases where the multiplier t has not necessarily perfectly uniform distribution. The generalization used the classical notion of discrepancy [7, 17, 27]. Recall that the discrepancy D(#) of an N element sequence # = # 1 , #N of elements of the interval [0, 1] is defined as D(#) sup J#[0,1] A(J, N) N J , where the supremum is extended over all subintervals J of [0, 1] J is the length of J , and A(J, N) ....

R. Kuipers and H. Niederreiter. Uniform Distribution of Sequences. WileyInterscience, NY, 1974.

....fraction (v u) of the n indices; i.e. the members of the sequence fall in a fair manner. We sometimes consider a weaker condition that (fa n g) be merely dense in [0; 1) noting that equidistributed implies dense. Armed with the above nomenclature, we paraphrase from [3] and references [27] [21] 33] 25] in the form of a collective de nition: De nition 2.1 (Collection) The following pertain to real numbers and sequences of real numbers ( n 2 [0; 1) n = 0; 1; 2; For any base b = 2; 3; 4 : we assume, as enunciated above, a unique base b expansion of whatever real ....

....2. If, for some b, is b dense then is irrational. Proof. The base b expansion of any rational is ultimately periodic, which means some nite digit strings never appear. 3. Almost all real numbers in [0; 1) are absolutely normal (the set of non absolutelynormal numbers is null) Proof. See [27], p. 71, Corollary 8.2, 21] 4. is b dense i the sequence (fb n g) is dense. Proof. See [3] 5. is b normal i the sequence (fb n g) is equidistributed. Proof. See [27] p. 70, Theorem 8.1. 6. Let m 6= k. Then is b k normal i is b m normal. Proof. See [27] p. 72, ....

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L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, WileyInterscience, New York, 1974.

....finite digit string (d 1 d 2 : d k ) appears in ff. This is taken to be the limit as N 1 of the number of instances where ff j = d 1 , ff j 1 = d 2 , ff j k Gamma1 = d k (for j ranging from 1 to N 1 Gamma k) divided by N . We now introduce a standard definition from the literature [Kuipers and Niederreiter 1974, pp. 69, 71] Definition 2.1. A real number ff is said to be normal to base b if every finite string of k digits appears in the base b expansion of ff with well defined limiting frequency b Gammak . A number that is normal to every integer base b 2 is said to be absolutely normal. We ....

....of [0; 1) with a fair frequency, in the following exact sense: Definition 2.3. A sequence x in [0; 1) is said to be equidistributed if for any 0 c d 1 we have lim N 1 C(x; c; d; N) N = d Gamma c: This definition is identical to that of uniform distribution modulo 1 , as given in [Kuipers and Niederreiter 1974, p. 1] In our development we shall need one (out of several) existing theorems on equidistribution, namely the following [Kuipers and Niederreiter 1974, p. 3] where we have added the simple extension that covers the weaker condition of density along with equidistribution: Theorem 2.4. Let (x ....

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L. Kuipers and H. Niederreiter, Uniform distribution of sequences, WileyInterscience, New York, 1974.

....) in [0; 1) d is called uniformly distributed modulo 1 (u.d. if lim N 1 DN (x n ) 0: Equivalently the sequence of (weighted) counting measures 1 N N 1 X n=0 xn (with x (E) E (x) converges weakly to the Lebesgue measure. For more details on this subject we refer to the monographs [5, 8]. 1 This work was supported by the Austrian Science Foundation, grant Nr. S 8302 MAT. 2 This work was supported by the Austrian Science Foundation, grant Nr. P 12441 MAT and grant Nr. S 8304 MAT. 1 Let q 1 be an integer. Then any non negative integer n has a unique q ary digital ....

....[g 0 ; g 1 ; g r ] be the continued fraction expansion of p q . Then for all integers v; w with 0 v v w q we have v w 1 X k=v fk p q g w 2 4 (g 1 g r ) Proof. This follows immediately by applying Koksma s inequality (Theorem 5. 1 in [8], Chapter 2) for f(x) x and by formula (3.18) in [8] Chapter 2, concerning the discrepancy of the point set fk p q g; k = v; v 1; v w 1. 2 Lemma 7 Let q; n 1 be integers. Let S 1 and and be positive reals with 0 ; S. Let y 0 ; y 1 ; be a sequence in [0; ....

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L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences , Wiley, 1974.

....the strip. This implies that vertices x which project into U are the origin of a translation of F which is also contained in S. There are infinitely many such lattice points, and so each finite subpattern of Q # occurs infinitely often in Q # . Since L projects with a 10 uniform density onto K [KNi], the relative number of vertices which are the origin of a subpattern F is equal to the ratio (K # U) K # (# # C) where is the m # dimensional Lebesque measure. Next we compare two patterns Q # and Q # # . If ## # # # = ### # K, then there exists a sequence of patterns Q # ....

....original acceptance region. We therefore can construct a subpattern which is a product of 1d quasicrystals obtained from 2d lattices. The latter is known to be relatively dense. In fact, in a 1d quasicrystal obtained from a 2d lattice there occur only two, in sigular cases three di#erent intervals [KNi]. This theorem makes the connection to the theory of almost periodic functions of Bohr (see e.g. Bes] According to Bohr, a function f(x) is uniformly almost periodic i# for each # 0 there exists a relatively dense set of vectors x i such that f(x) f(x x i ) # for all i, ....

L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, WileyInterscience: New York 1974.

.... 2 fStd[D 2 (P ) g 2 ; so p Ef[D 2 (P ) 2 g is an upper bound on both the mean and the standard deviation of D 2 (P ) Thus, by Theorem 5 E[D 2 (P ) O(N Gamma1 (log(N) s Gamma1) 2 ) which is the same asymptotic order as a lower bound on D 2S (P ) derived by Roth [1954] Kuipers and Niederreiter [1974, p.102] refined this lower bound to give an explicit constant (see also [Niederreiter 1978, Equation (3.10) and p. 972] D 2;S (P ) B(s)N Gamma1 [log(N ) s Gamma1) 2 where B(s) 12 Gamma1=2 for s = 1 f16 s [ s Gamma 1) log(2) s Gamma1) 2 g Gamma1 for s 2 A lower ....

Kuipers, L. and Niederreiter, H. 1974. Uniform Distribution of Sequences. John Wiley, New York.

....between 0 and 1 of their cumulative distribution functions that distinguishes distributions. There are many statistics to measure the overall di erence between two cumulative distributions. We have chosen a variant of the generally accepted Kolmogorov Smirnov (K S) test, namely Kuipers statistic [10], which is the sum of the maximum distances of Sn (x) above and below PU (x) Vn = D D = max 0 x 1 [S n (x) PU (x) max 0 x 1 [P U (x) Sn (x) 1) The method is demonstrated in Figure 4a. This statistic guarantees equal sensitivities at all values of x, in contrast to the original K S ....

L. Kuipers and H. Niederreiter. Uniform Distribution of Sequences. Wiley, New York, 1974.

....= jfn N : fx n g 2 Igj. The discrepancy (at the N th stage) is DN (x) sup 0 a b 1 AN ( a; b) N (b a) As one would expect, DN (x) 0 as N 1 precisely when x is uniformly distributed. For a proof of this (and other statements in this section) see Kuipers and Niederreiter [5]. We will nd it useful to consider a slightly restricted notion of discrepancy: D N (x) sup 0 a 1 AN ( 0; a) N a : This is equivalent (in the sense that D N DN 2D N ) but has the advantage of having a more convenient expression: if we order (fx n gg N n=1 as y ....

Kuipers, L. and Niederreiter, H., Uniform distribution of sequences, John Wiley and Sons, New York, 1974.

....random walk on the circle S 1 taking steps # for irrational #. He bounds the rate of convergence to stationarity giving results that depend on the degree of irrationality of #. The bounds use standard tools from uniform distribution mod(1) Leveque s inequality and the Erdos Turan bound [25]. In the notation of this section, Leveque s inequality on S 1 gives D(K # x , m) # C# 2# 3 for a universal constant C. This also follows from Theorem 4.2. The Erdos Turan bound gives D(K # x , m) # C( 1 h # # log h) for any positive integer h. Optimizing in h gives as a slight ....

Kuipers, L. and Niederreiter, H. (1974), Uniform Distribution of Sequences, Wiley, N.Y.

....5. Let ff be a quadratic irrational (or an irrational number with bounded partial quotients) Let ffi 0 and = ffi=3. Let H be a fixed positive constant larger than i(1 ) where i denotes the Riemann zeta function. Let k be a positive integer. By the Erdos Tur an inequality, we know (cf. [4], example 3.2 p. 124) # ( n 2 [M 1; M N ] fi fi fi fi fi 1 H(k 1) 1 fffng 1 Hk 1 ) 1 )N Hk 2 Gamma C log 2 N; where fug denotes the fractional part of the real number u. Thus there exists N 0 not depending on M such that if N maxfN 0 ; k 2 2 g there exists ....

L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences (John Wiley, New York, 1974).

....such that : ffl y 0 2 V Gamma x (ffl) so f(y 0 ) f(x) ffl y 0 y, so f(y 0 ) f(y) Hence, we have f(x) f(y) 5 1. 3 Isotropic Volume Discrepancy We will now introduce the notion of isotropic volume discrepancy of sequences which looks like the usual isotropic discrepancy (cf. [4] et [5] Denition 1.11 The Isotropic Volume Discrepancy of the rst N points of a sequence oe = x n ) n1 is dened by : DJN (oe) sup C2Cs EVN (oe; C) where C s is the family of all convex sets included in I s . Theorem 1.12 For all sequence oe = x n ) n1 of elements of I s , we have ....

....that (O) P ffl ) ffl and Cardfx n 2 O oe N g = Cardfx n 2 P ffl oe N g: Thus we have : EVO (oe N ) O) Gamma VO (oe N ) P ffl ) Gamma V P ffl (oe N ) ffl = EV P ffl (oe N ) ffl: Hence DJN (oe) sup P2Fs EVN (oe; P ) Proof : of Theorem 1.12. We use the same method as in [4] (p. 94 97) to estimate the classical isotropic discrepancy. By the above lemma, it suOEces to estimate EVN (oe; P ) for a P 2 F s . To simplify this estimation we will construct a subset of P , say P 1 easier to handle than P . Since P 1 ae P : EV P (oe N ) P ) Gamma V P (oe N ) P 1 ) ....

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L.Kuipers and H.Niederreiter, Uniform distribution of sequences, Wiley, New York, 1974.

....distribution of the orientations that as you add more orientations to your list the frequencies B and B 0 , of hits in any two open intervals B and B 0 of equal size, have the same limit. There is a well known Weyl criterion for proving uniform distribution: Theorem 4. 5 (Weyl) [KuN]. fx j g 2 SO(d) is uniformly distributed if and only if for every continuous nontrivial irreducible representation f of SO(d) lim N 1 (1=N) N X j=1 f(x j ) 0 4:11) This criterion is best known in the special case d = 2 (which we need for the pinwheel) where it says that a sequence fx j ....

L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, John Wiley, New York, 1974.

.... of x 1 ; xN is de ned by DN (x n ) sup x2[0;1] k j N (x n ; 0; x) j (2) and the L 2 discrepancy D (2) N by D (2) N (x n ) Z [0;1] k ( N (x n ; 0; x) 2 dx 1 2 : 3) 1 (Sometimes 1 N sup j N (x n ; 0; x) j is called discrepancy and is also denoted by DN [5, 3]. Furthermore observe that the value of DN resp. that of D (2) N will not change if we use closed boxes [0; x] instead of the half open boxes [0; x) Roth [7] proved that there are constants c k 0 such that DN (x n ) D (2) N (x n ) c k (log N) 1 2 (k 1) 4) In [1] these lower ....

.... only depends on k. This result is worse than Ruzsa s [8] in the case k = 2 but it seems to be the rst quantitative relation between DN (x n ; Q) and DN (x n ) in arbitrary dimensions which is better than the quantitative relation between the isotropic discrepancy JN (x n ) and DN (x n ) see [5]) In section 3 we will generalize (4) to discrepancies with respect to arbitrary simple convex polytopes P . A k dimensional polytope is simple if every vertex of P has exactly k adjacent ones. Similarly to (5) and (6) we can de ne discrepancies DN (x n ; P) sup y2[0;1) k ;0 1 j N (x n ....

L. Kuipers and H. Niederreiter, \Uniform Distribution of Sequences", John Wiley and Sons, London, 1974.

....every hexadecimal string appears with proper frequency) then the number is also normal to base 2, or for that matter to any power oftwo base. The wording of this latter part is critical: there exist numbers normal to some base b but not to some other base a that is not a rational power of b [9, 15]. For example, the standard Cantor set has members that are normal to base 2, yet none of its members is normal to base 3. Moreover, there are results on the class of absolutely abnormal numbers, meaning numbers not normal to any base. Any rational number is of this class, of course, yet the ....

L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, WileyInterscience, New York, 1974.

....m X j=1 j 2 f(1; y j )j with respect to all possible partitions 0 x 1 x 2 xn 1; 0 y 1 y 2 ym 1: of I 2 . The shortest route to the results makes use of concepts such as uniform distribution, discrepancy, and numerical integration. The key reference is [9]. The starting point is the inequality Z I F (x)dx 1 N N X i=1 F (x i ) DN V (F ) where DN denotes the discrepancy of the sequence x 1 ; x 2 ; xN and V (F ) is the total variation of the function F (which is assumed to be of bounded variation in the sense of Hardy and Krause) ....

L. Kuipers and H. Niederreiter. Uniform Distribution of Sequences. John Wiley & Sons, New York, 1974.

....period, then Eq. 2.1) is a partial period exponential sum. Examining Eq. 2.1) shows it to be a sum of k quantities on the unit circle. A trivial upper bound is thus jC(k)j k. If the sequence fxn g is indeed uniformly distributed, then we would expect jC(k)j = O( p k) Kuipers and Niederreiter [13]. Thus the desire is to show that exponential sums of interest are neither too big nor too small to reassure us that the sequence in question is theoretically equidistributed. Since we are interested in studying sequences for use in parallel, we must consider the cross correlations among the ....

L. Kuipers and H. Niederreiter, Uniform distribution of sequences, John Wiley and Sons: New York, 1974.

....N n=1 1B ( n) and let B be a family of sets of form Q s i=1 [0; u i [ with u = u 1 ; Delta Delta Delta ; u s ) 2 [0; 1[ s . The discrepancy of the N first terms of P is defined by D N (P) sup B fflB j AN (B; P) N Gamma s (B)j: The two following propositions are equivalent [10]: 1. P is uniformly distributed on [0; 1] s . 2. lim N 1 D N (P) 0. To be able to bound the approximation error in the computation of the integral R [0;1] s f(u)du, we have to define the variation in the sense of Vitali and the variation in the sense of Hardy and Krause. Let P set of ....

L. Kuipers and H. Niederreiter. -- Uniform Distribution of Sequences. -- John Wiley, New York, 1974.

.... (x n ) The discrepancy in space L p of the N first terms of P is defined by T (p) N (P) Z [0;1] s fi fi fi fi fi AN (B; P) N Gamma s (B) fi fi fi fi fi p dz 1=p : Irisa Improvement of Halton sequences distribution 5 The four following propositions are equivalent [8]: 1. P is uniformly distributed on [0; 1] s : for each subinterval B of [0; 1[ s ; 1 N N X n=1 1B (x n ) Gamma N 1 s (B) 2. lim N 1 D N (P) 0. 3. lim N 1 DN (P) 0. 4. lim N 1 T (p) N (P) 0. To be able to bound the approximation error in the computation of the ....

L. Kuipers and H. Niederreiter. -- Uniform Distribution of Sequences. -- John Wiley, New York, 1974.

....4 THE MULTIDIMENSIONAL PROBLEM In this section we sketch the solution to the multidimensional problem for radial basis neural networks. The statement and proofs of the results depend on certain number theoretic results concerning uniform distribution, discrepancy, and numerical integration [8]. We start with the following inequality fi fi fi fi fi Z I F (x)dx Gamma 1 N N X i=1 F (x i ) fi fi fi fi fi DN V (F ) where DN denotes the discrepancy of the sequence x 1 ; x 2 ; xN and V (F ) is the total variation of the function F 2 BV (which is assumed to be of bounded ....

.... fi fi fi fi fi Z I F (x)dx Gamma 1 N N X i=1 F (x i ) fi fi fi fi fi DN V (F ) where DN denotes the discrepancy of the sequence x 1 ; x 2 ; xN and V (F ) is the total variation of the function F 2 BV (which is assumed to be of bounded variation in the sense of Hardy and Krause [8]) Note that x 2 R n , as well as each x i . Standard results in the estimation of the discrepancy DN show that DN log n Gamma1 N N if the sequence x i is a good lattice set. It is possible to show that the variation in the sense of Hardy and Krause of a product of two functions V (fg) is ....

L. Kuipers and H. Niederreiter. Uniform Distribution of Sequences. John Wiley & Sons, New York, 1974.

....1 j . We need p(fi; q) to be less than a constant strictly less than 1. We rewrite p(fi; q) as: p(fi; q) 1 Gamma Pr i q=2 1 j bfitc q q Gamma q=2 1 j j = 1 Gamma Pr bfitc q q 2 1 2 1 j ; 1 Gamma 1 2 1 j : This suggests to use the classical notion of discrepancy [8, 14, 21]. Recall that the discrepancy D( Gamma) of an N element sequence Gamma = ffl 1 ; fl N g of elements of the interval [0; 1] is defined as D( Gamma) sup J [0;1] fi fi fi fi A(J; N) N Gamma jJ j fi fi fi fi ; where the supremum is extended over all subintervals J of [0; 1] jJ j ....

R. Kuipers and H. Niederreiter. Uniform distribution of sequences. WileyInterscience, NY, 1974.

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L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley, New York, 1974.

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L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, John Wiley, New York, 1974.

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H. Niederreiter and L. Kuipers. Uniform Distribution of Sequences. John Wiley and Sons, New York, 1974.

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L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences,Wiley,NewYork, 1974.

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L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences. New York: Wiley, 1974.

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L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences. New York: Wiley, 1974.

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L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, 1974; MR0419394 (54 #7415).

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Kuipers, L. and Niederreiter, H., Uniform distribution of sequences, John Wiley and Sons, New York, 1974.

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L. Kuipers and H. Niederreiter. Uniform Distribution of Sequences. Wiley-Interscience, New York, 1974.

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L. Kuipers and H. Niederreiter, Uniform distribution of sequences, J. Wiley and Sons, 1974.

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L. Kuipers and H. Niederreiter, Uniform distribution of sequences, John Wiley & Sons, New York, 1974.

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L. Kuipers and H. Niederreiter, Uniform distribution of sequences, John Wiley & Sons, New York, 1974.

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Kuipers, L. and Niederreiter, H. Uniform Distribution of Sequences. Wiley, New York, 1974.

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L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-LondonSydney, 1974.

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L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-LondonSydney, 1974.

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L. Kuipers, H. Niederreiter (1974), Uniform Distribution of Sequences, John Wiley & Sons, New York

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L. Kuipers and H. Niederreiter. Uniform Distribution of Sequences. John Wiley, New York, 1974.

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L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences. John Wiley & Sons, Inc., N.Y., 1974.

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L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York (1974).

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L. Kuipers, H. Niederreiter, Uniform distribution of sequences. Wiley (1974).

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L. Kuipers and H. Niederrieter, Uniform distribution of sequences, Wiley 1974.

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