We analyzed a new method for proving equidistribution of numbers. The proposed method is simple and can be used to prove equidistribution of all known classes of numbers such
as Weyl\u27s numbers and Koksma\u27s numbers. Emphasis of this approach is put on the complete equidistribution of numbers and a non-trivial result in this direction is obtained

N. Limić

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Mathematical Communications 7(2002), 103-111 103On completely equidistributed numbers∗Nedzˇad Limic´†Abstract. We analyzed a new method for proving equidistribu-tion of numbers. The proposed method is simple and can be used toprove equidistribution of all known classes of numbers such as Weyl’snumbers and Koksma’s numbers. Emphasis of this approach is put onthe complete equidistribution of numbers and a non-trivial result in thisdirection is obtained.Key words: equidistributed numbers, pseudo-random numbersAMS subject classifications: 11K45, 65C10Received March 14, 2002 Accepted October 9, 20021. IntroductionPoints of the cube D = [0, 1]d are denoted by βk. A denumerable sequence offunctions xk : G → Rd, where G ⊂ Rr, defines a denumerable sequence of pointsβk ∈ D, by the expression:βk = xk(t) [mod 1], t ∈ G.The set G is called the set of seeds. Let 1S be the indicator of a set S =∏j [aj , bj] ⊂D, 0 ≤ aj < bj ≤ 1. We say that a sequence β = {βk : k ∈ N} is equidistributed inD in Weyl’s sense iflimN→∞1NN∑k=11S(βk) = mess (S)for each S ⊂ D.Let I be a bounded interval of R. Familiar examples of functions xk : I → R,defining equidistributed sequences in [0, 1], are as follows. The sequence of functions,xk(t) = np t, t ∈ I = (0, 1), (1)∗This work was partially supported by grant 037011 from the Ministry of Science and Technol-ogy, Republic of Croatia†Department of Mathematics, University of Zagreb, Bijenicˇka 30, HR-10 000 Zagreb, Croatia,e-mail: nlimic@math.hr104 N.Limic´where p ∈ N, defines Weyl’s numbers βk ∈ [0, 1] of parameter p. The sequence offunctionsxk(t) = Mk t, t ∈ I = (0, 1), (2)defines multiplicatively generated numbers in [0, 1] for each M ∈ {2, 3, . . . , }. Thesequence of functionsxk(t) = tk, t ∈ I = (1, a), (3)with any a > 1, defines Koksma’s numbers in [0, 1]. All these numbers are equidis-tributed for almost all seeds [We, F1, Ko] (see also [Ca]). This means that thereexists a set T ⊂ I,mess (T ) = mess (I), such that βk = xk(t) [mod 1] is equidistrib-uted in [0, 1] for t ∈ T .The functions (1) - (3) can be used to generate points βk = (βk1, βk2, . . . , βkd)in D = [0, 1]d by two different possibilities:βkj = xj+(k−1)d(tj) [mod 1], t = (t1, t2, . . . , td) ∈ Id, (4)βkj = xk+j−1(t) [mod 1], t ∈ I. (5)We say that the numbers {βk : k ∈ N} ⊂ [0, 1] are d-dimensionally equidistributedin [0, 1] if the sequence of points βk defined by (5) is equidistributed in [0, 1]d for aset T ⊂ I,mess (T ) = mess (I). In case this property holds for any d > 1, we saythat the sequence {βk : k ∈ N} is completely equidistributed. Koksma’s numbersare completely equidistributed [F2].The objective of this work is to present a unique method by which the encoun-tered results can be obtained. This method is an alternative to the known ones[We, F1, F2, Ko]. It is based on the Weyl’s criterion and a.e. convergence of asequence of measurable functions. Hence, this new method is based on a tool whichis extensively used in the theory of probability. The proposed method is very simplefor the case of points βk which are defined in terms of linear functions xk. Thisincludes numbers defined by the functions (1), (2), and p-dimensionally equidistri-bution of numbers defined by (1). The method is described in Section 2. Nonlinearfunctions xk are considered in Section 3, where the complete equidistribution ofKoksma’s numbers is proved by the proposed method. In addition, to illustratean efficiency of the method, a new class of completely equidistributed numbers ispresented at the end of the section. Concluding remarks are given in Section 4..2. Linear caseLet Zd be the set of multi-indices m = (m1,m2, . . . ,md),mk ∈ Z. Let {xk : k ∈ N}be a sequence of linear functions from I = [0, 1] to R and let points βk ∈ D bedefined by (4). A brief notation is βk = xk(t) [mod 1].In accordance with Weyl criterion we consider the quantitiesWN (β,m) =1NN∑k=1exp(2πim · βk),where a ·b is the scalar product of a,b ∈ Rd. Now we use the fact that βk ∈ D areequidistributed iff WN (β,m) → 0 for N → ∞ and each m = 0.On completely equidistributed numbers 105Let us use the notation z → e(m, z) = exp(2πim · z) and let xk(·) be definedby (1), (2), and (4). Then the sequence of functions t → e(m,xk(t)) defines anorthonormal sequence of elements in L2(Id).The functionsD  t → fN (t) = 1NN∑k=1e(m,xk(t)), (6)m is fixed, are the main object in our proof of equidistribution. Apparently, theirL2(D)-norms are ‖fN‖2 = 1/√N and certain subsequences converge a.e. to zeroon D. Hence, certain subsequence of WN (β,m) converges to zero. In order topass from a subsequence to the whole sequence {WN (β,m) : N ∈ N}, we need anauxiliary result [Do, Ha]:If α > 1, then the sequenceI(α) = {N(1), N(2), . . . , N(k), . . . }, N(k) ∈ [kα, kα + 1)is an infinite subsequence of N.Lemma 1. Let for each m = 0 there exist α(m) > 1 such that a sequence ofpoints {βk : k ∈ N} ⊂ D satisfies the conditionslimN∈I(α(m))WN (β,m) = 0, m ∈ Zd, m = 0.Then {βk : k ∈ N} is equidistributed in D.Now we can prove the main result of this section.Proposition 1. Let a sequence of linear functions xj : [0, 1] → R define pointsβk ∈ D by (4). If for each m ∈ Zd,m = 0, the sequence e(m,xk) ∈ L2(D) isorthonormal, (e(m,xk) | e(m,xl))= δkl,then the sequence of points βk is equidistributed in D.Proof. From Chebyshev’s inequalitymess(∣∣fN (t)∣∣ > ε) ≤ ‖fN‖22ε2=1ε2 N,one concludes that fN (t) → 0 for almost all seeds t ∈ D if N → ∞, N ∈ I(α).Then the statement follows from Lemma1. ✷Example 1. Let qj , j ∈ N, be real numbers, xj(t) = jp t+ qj, and let xk : D →Rd be defined by components xj+(k−1)d(tj), j = 1, 2, . . . , d. Then the points (4) areequidistributed in D for almost all seeds t ∈ D.Example 2. Let d pairs (Mj , qj), Mj ∈ {2, 3, . . .}, qj ∈ R, define the functionsxk : [0, 1] → R by:tj → xj+(k−1)d(tj) = Mkj tj + qj , j = 1, 2, . . . , d, k ∈ N.Columns with elements Mkj tj + qj , j = 1, 2, . . . , d, are denoted by Mkt + q. Toshow that the sequence of points βk = xk(t) [mod 1] is equidistributed in D foralmost all seeds t ∈ D we need an auxiliary result.106 N.Limic´Let Mj ∈ {2, 3, . . .}, j = 1, 2, . . . , d, and q,σ ∈ Rd with components qj , σjrelated by the equations σj(Mj − 1) + qj = 0. Two sequences of points in D:βk+1 = Mβk [mod 1], β0 ∈ D,γk+1 = Mγk + q [mod 1], γ0 = β0 + σ [mod 1],are simultaneously either equidistributed or not. This statement can be easily proved.By a straightforward calculation one gets γk = βk + σ [mod 1], k ∈ N, and thestatement follows from WN (γ,m) = exp(2πim · ρ)WN (β,m).Example 3. Let us define βj = jpt + qj where qk ∈ R. For each m ∈ Zp,m = 0, the functions:xk(t) = tp∑j=1mj (k + j − 1)p + zk,where zk =∑j mjqj, fulfil the conditions of Proposition 1. This means that γk =xk(t) [mod 1] are equidistributed in [0,1] for almost all seeds. Thus, βj are p-dimensionaly equidistributed in [0, 1] for almost all seeds.3. Nonlinear caseA function f on Rd is said to be piecewise continuous if there exists a finite col-lection of disjoint subsets Sk ⊂ Rd, k = 1, 2, . . . , L, Rd = ∪kSk, and bounded,uniformly continuous functions on Rd, f1, f2, . . . , fL, such that f =∑Lj=1 fj1Sj . Afunction fD on D ⊂ Rd is said to be piecewise continuous if there exists a piecewisecontinuous f on Rd such that fD = f |D.Let {βk : k ∈ N} ⊂ D and L be a subspace of the linear space of piecewisecontinuous functions from D into R such that the equality1NN∑k=1f(βk) =∫Df(x) dx, f ∈ L, (7)is valid iff {βk : k ∈ N} is equidistributed in D. For instance, L can be the linearspace of continuous functions on D, the linear space spanned by indicators of cubesin D, spanned by piecewise continuous functions on D.An orhonormal basis {em,m ∈ N0} in L2(D), such that em are piecewise con-tinuous on D, e0(x) = 1, is denoted by E . Let G ⊂ Rr be a closed, bounded setand let a sequence of piecewise continuous functions xk : G → D, k ∈ N, be given.Instead of functions (6) we consider nowfN (t) =1NN∑k=1em(xk(t)). (8)Each em is bounded on D so that Lemma1 has a straightforward generalization:Corollary 1. Let m ∈ N be fixed and for some α > 1:limN∈I(α)fN (t) = 0 a.e. on G.On completely equidistributed numbers 107Then the sequence {fN : N ∈ N} converges to zero a.e. on G.In the previous sections the quantitiesK(k, l) =(em(xk)∣∣ em(xl) ),had values δkl. In the present generalization they may have nonzero values for k = l.Theorem 1. Let the orthonormal basis E ⊂ L2(D), and a sequence of piecewisecontinuous functions xk(·), from a bounded closed subset G ⊂ Rr into D, have thefollowing properties:(i) For each m ∈ N there exist two positive numbers κ(m), ρ(m) such that∣∣K(k, l)∣∣ ≤ κ(m) |k − l|−ρ(m);(ii) Let K be the linear space spanned by elements of E. For each f ∈ L and ε > 0there exist f−, f+ ∈ K such thatf−(x) ≤ f(x) ≤ f+(x), x ∈ D,‖f+ − f−‖1 < ε.Then the sequence of points βk = xk(t) is equidistributed in D for almost allt ∈ G.Proof. The functions (8) have their L2(D)-norms:‖fN‖2 ≤ 1N2N∑k,l=1∣∣K(k, l)∣∣.Because of (i), the right-hand side can be bounded from above by a number ωN−γ , γ >0 [Ha], where ω depends on m and does not depend on N . Thus, any subsequenceof functions {fN(·) : N ∈ I(α)}, with an α such that αγ > 1, converges a.e. to zeroon G. By Corollary 1 the sequence {fN(·) : N ∈ N} converges to zero for almost allt ∈ G.Let us considerf = lim supN1NN∑k=1f(βk), f = lim infN1NN∑k=1f(βk).Because of (ii) f − f ≤ ‖f+ − f−‖1 < ε, implying the equidistribution of numbersβk. ✷Example 4. Koksma’s numbers, defined by functions (3), can be considered nowby specifying G = [1, a], em(x) = exp(±2πimx) and L = C([0, 1]). In accordancewith (i) of Theorem1 we consider the functions t → g(m, k, l, t) = 2πm(tk− tl), k >l, and integralsI(m, k, l) =∫ a1{sincos} (g(m, k, l, t))dt.108 N.Limic´Numbers K(k, l) of Theorem 1 are linear combinations of I(m, k, l). It suffices toconsider the case with sin-function. The quantity I(m, k, l) is a finite series of termsobtained by dividing the interval [1, a] into subintervals on which the integrand doesnot change its sign. Terms of such obtained series have strictly decresing magnitudesand alternating signs. Hence, the value of I(m, k, l) may be estimated by the firstterm:I(m, k, l) <∫ 1+τ1sin(g(m, k, l, t))dt,where 1, 1+τ are the first two zeros of the integrand, i.e. g(m, k, l, 1) = g(m, k, l, 1+τ) = 0. It is easy to check τ < 1/(2m(k − l)). Hence, Theorem1 can be applied.In the proof of equidistribution of the previous example two properties of thefunctions g(m, k, l),m ∈ N, are used.Corollary 2. Let xk : I → R, and g(k, l, t) = xk(t) − xl(t), have the followingproperties:P1) For sufficiently large k, l, the functions g(k, l), g′(k, l) are continuous, positiveand strictly increasing on I;P2) For sufficiently large k, l, there exist positive numbers c1, c2 such thatg′(k, l) ≥ c1 |k − l|c2 , k, l ∈ N.Then the sequence of numbers βk = xk(t) [mod 1] is equidistributed in [0, 1] foralmost all seeds t ∈ I.Example 5. The complete equidistribution of Koksma’s numbers follows fromtheir d-dimensional equidistribution for each d ∈ N. The corresponding set of seedsis T = ∩Td. Let m ∈ Zd,m = 0, andg(m, k, l, t) = 2πd∑j=1mj(tk+j−1 − tl+j−1) = p(t) (tk − tl),where p is a polynomial of degree d−1 with coefficients depending on m. Zeros of p,t1, t2, . . . , td−1, are surrounded by intervals Ir = (−ε+ tr, tr + ε), where ε = k−1/d.There are disjoint closed intervals Jr such that J = [1, a] \(∪r Ir) = ∪rJr. Hence,∣∣∣I(m, k, l) − ∑r∫Jrsin(g(m, k, l, t))dt∣∣∣ < ∑rmess (Ir) < 2(d− 1)k−1/d.It suffices to demonstrate that the functions g(m, k, l) (or −g(m, k, l)) on each Jrhave the properties P1), P2) of Corollary 2. From the expressiong′(m, k, l, t) =p(t)t(k tk − l tl)[1 +tp′(t)p(t)tk − tlk tk − l tl],we have the following estimate:supt∈Jr∣∣g′(m, k, l, t)∣∣ ≥ c0(m, d) (k − l)1/d[1 − O(1k − l)].On completely equidistributed numbers 109Thus, for sufficiently large k− l we have |g′(m, k, l, t)| > c1(m, d)(k− l)1/d, i.e.the properties P1), P2) are valid either for g(m, k, l) or −g(m, k, l) .Now we present a class of functions xk, generating completely equidistributednumbers, and generalizing Koksma’s numbers defined in terms of functions (3),xk(t) = exp(k ln t). One expects that the functions xk(t) = exp(kρ ln t) on (1,∞)also generate completely equidistributed numbers. First, the functions exp(kρ ln t), ρ >1, are replaced by a more general sequence of functions uk(·)w(k).Theorem 2. Letxk(t) = uk(t)w(k), t ∈ I, k ∈ N,where(i) w is strictly increasing from N into [b,∞), b > 0, with the following properties;w(k + r) ≤ φ+ w(max{k, r}),w(k + r) − w(k) ≥ φ− w(k)α w(r)β , k, r ∈ N,for some positive numbers φ±, α, β.(ii) uk, u′k, k ∈ N, are continuous, strictly increasing from a closed interval I into[a,∞), a > 1, with the following additional properties:uk+1uk≥ 1, c0 ≤u′k−1(t)uk−1(t)≤ u′k(t)uk(t)≤ aγw(k)δ ,for some c0 > 0 and γ ≥ 0, δ ≥ 0, δ < α.Then the functions xk generate numbers βk = xk(t) [mod 1], completly equidis-tributed in [0, 1] for almost all t ∈ I.Proof. It can be assumed w(1) = 1. The functions gk = xk−xl fulfil conditionsP1), P2) of Corollary 2, so that the numbers βk are equidistributed for almost allt ∈ I. It has to be proved that they are d-dimensionaly equidistributed for all d > 1.Let m ∈ Zd and the corresponding functiong(m, k, l, t) =d∑j=1mj(xk+j−1(t) − xl+j−1(t)).It suffices to prove that g(m, k, l) satisfy P1) and P2) for sufficiently large k − l.Let us define the functionshk(m, t) =d∑j=1mjxk+j−1(t)xk+d−1(t).Theng(m, k, l, t) = hk(m, t)xk+d−1(t) − hl(m, t)xl+d−1(t).From properties (i), (ii), we have∣∣∣∣xk+j−1(t)xk+d−1(t)∣∣∣∣ ≤ a−φ−w(k−j+1)α ≤ a−φ−w(k)α ,110 N.Limic´∣∣∣∣∣(xk+j−1(t)xk+d−1(t))′∣∣∣∣∣ ≤xk+j−1(t)xk+d−1(t)u′k+d−1(t)uk+d−1(t)w(k + d− 1) ≤w(k + d− 1) a−φ−w(k)α+γφδ+w(k)δ .Hence, the functions xk+j−1/xk+d−1, j < d, and their derivatives tend to zero on Ifaster than any power of w(k) as k increases. We can assume md = 0, as otherwisethe problem is reduced to (d − 1)-dimensional equidistribution. Since the sign ofg(m, k, l) is irrelevant, we assume md > 0. The functions hk(m) tend to md on theset I:supt∈I∣∣∣hk(m, t) − md∣∣∣ ≤ κ1(m) a−φ− w(k)α .Similarly,supt∈I∣∣∣h′k(m, t)∣∣∣ ≤ κ2(m)w(k + d− 1) a−φ−w(k)α+γφδ+w(k)δ .The functions g(m, k, l) seem to behave like the functions g0(k, l, t) = md[xk+d−1(t)−xl+d−1(t)]. Fromg′0(k, l, t) =md[u′k+d−1(t)uk+d−1(t)w(k + d− 1)xk+d−1(t) −u′l+d−1(t)ul+d−1(t)w(l + d− 1)xl+d−1(t)]it followsg′0(k, l, t) ≥ c0 md[w(k + d− 1) − w(l + d− 1)]xk+d−1(t) ≥ c0 md φ− (k − l)β,as well asg′0(k, l, t) ≥ md x′k+d−1(t)[1 − xl+d−1(t)xk+d−1(t)]≥ mdc0 xk+d−1(t)[1 − xl+d−1(t)xk+d−1(t)].Therefore, for each 1/2 > ε > 0 there exist κ(m, ε) > 0, n(ε) ∈ N, such that thevalue of square brackets is larger than 1/2 when k > k − l > n(ε).∣∣g′(m, k, l, t) − g′0(k, l, t)∣∣ ≤ κ(m, ε) g′0(k, l, t)w(k + d− 1) a−φ−w(k)α+γφδ+w(k)δ .Thus the sequence of functions g(m, k, l) has properties P1), P2) of Corollary 2 forsufficiently large k − l. ✷4. DiscussionTheorem 2 generalizes results about the functions xk of (3) by replacing powers tkwith tw(k) where w increases faster than the linear function of k. A similar resultexists in which w increases slower than the linear function such as w(k) = kρ, ρ <1. Classes of completeley equidistributed numbers are important in descriptionof mathematical samples of infinite length of the uniformly distributed randomvariable. Therefore, Koksma’s numbers as well as classes of Theorem 2 provide uswith non-trivial examples of such samples and Monte Carlo simulated realizations.Properties P1), P2) of Corollary 2 seem to be completely analogous to Propertiesof Satz 3. in Koksma’s article [Ko]. Differences are rather technical.On completely equidistributed numbers 111References[Ca] J.W. S. Cassels, An Introduction to Diphantine Approximation, CambridgeUniversity Press, 1965[Do] J.Doob, Stochastic Processes, John Wiley, 1953[F1] J.N.Franklin, On the equidistribution of pseudo-random numbers,Quart.Appl.Math. 16(1958), 183–191.[F2] J.N.Franklin, Deterministic simulation of random processes, Math. Comp.17(1963), 28–59.[Ha] E. J.Hannan, Multiple Time Series, John Wiley, New York, 1970[Ko] J. F.Koksma, Ein mengentheoretischer Satz Uber die Gleichvertailung mod-ulo Eins, Compositio Math. 2(1935), 250–258.[We] H.Weyl, U¨ber die Gleichverteilung von Zahlen mod. Eins, Math. Ann.77(1916), 313–352.

On completely equidistributed numbers

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