Uniformly distributed sequences of partitions

Transcription

1 Uniformly distributed sequences of partitions Aljoša Volčič Università della Calabria Uniform distribution and QMC methods A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

2 α-refinement In 1976 S. Kakutani introduced the following nice geometric construction. Definition (α-refinement) If α ]0, 1[ and π {0 = t 0 < t 1 < < t k = 1} is any partition of [0, 1], the α-refinement of π (denoted by απ) is the partition obtained subdividing the interval(s) of π having maximal length in two parts in proportion α / 1 α. We can iterate the α-refinement. The α-refinement of απ will be denoted by α 2 π and for the successive α-refinements we will use the notation α n π, for n IN. If π = ω = {[0, 1]}, the trivial partition, the sequence of partitions {α n ω} is called the Kakutani sequence of partitions of [0, 1] of parameter α. A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

3 α - Kakutani-sequence {κ n } for α = 3/4 ω A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

4 α - Kakutani-sequence {κ n } for α = 3/4 ω α ω α 1- α A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

5 α - Kakutani-sequence {κ n } for α = 3/4 ω α ω α 1- α α 2 ω α 2 α (1- α) 1- α A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

6 α - Kakutani-sequence {κ n } for α = 3/4 ω α ω α 1- α α 2 ω α 2 α (1- α) 1- α α 3 ω α 3 α 2 (1- α) α (1- α) 1- α A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

7 α - Kakutani-sequence {κ n } for α = 3/4 ω α ω α 1- α α 2 ω α 2 α (1- α) 1- α α 3 ω α 3 α 2 (1- α) α (1- α) 1- α α 4 ω α 4 α 3 (1- α) α 2 (1- α) α (1- α) 1- α A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

8 α - Kakutani-sequence {κ n } for α = 3/4 ω α ω α 1- α α 2 ω α 2 α (1- α) 1- α α 3 ω α 3 α 2 (1- α) α (1- α) 1- α α 4 ω α 4 α 3 (1- α) α 2 (1- α) α (1- α) 1- α α 5 ω α 5 α 4 (1- α) α3 (1- α) α2 (1- α) α (1- α) 1- α A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

9 If α = 1/2, we get the well-known sequence of binary partitions. Definition (Uniform distribution) A sequence of partitions {π n }, π n {0 = t0 n < tn 1 < < tn k(n) = 1} is said to be uniformly ditributed (u.d.) if for any continuous real valued function f on [0, 1] we have lim n k(n) 1 k(n) i=1 f (t n i ) = 1 0 f (x) dx. Theorem (S. Kakutani) The sequence of partitions {α n ω} is uniformly distributed for any α ]0, 1[. A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

10 Few years after Kakutani s paper R.L. Adler and L. Flatto provided a simpler proof. Other authors, also in recent times, studied probabilistic variants of the construction (Lootgieter, van Zwet, Brennan & Durrett, Pyke & van Zwet). In the meanwhile the deterministic result has been almost forgotten, except for a paper in 1992 (F. Chersi, A.V.) which extended the notion of u.d sequence of partitions to complete separable metric spaces. In the last years Kakutani s idea has been revitalized putting it in a more general context and connecting it to other active areas of uniform distribution. A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

11 ρ-refinements Definition (ρ-refinement) Let ρ be a fixed finite non trivial partition of [0, 1] and π any partition of [0, 1]. The ρ-refinement of π (denoted by ρπ) is obtained splitting its interval(s) of maximal length directly homothetically to ρ. If ρ = {[0, α], [α, 1]} we get Kakutani s α-refinement. We can iterate the ρ-refinement. The ρ-refinement of ρπ will be denoted by ρ 2 π and for the successive ρ-refinements we will use the notation ρ n π, for n IN. Theorem (A.V. 2011) The sequence of partitions {ρ n ω} is uniformly distributed. When is {ρ n π} u.d.? ( C. Aistlaitner, M. Hofer). A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

12 ρ n ω with ρ = {[0, 1/2], [1/2, 3/4], [3/4, 1]} ω A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

13 ρ n ω with ρ = {[0, 1/2], [1/2, 3/4], [3/4, 1]} ω 1/2 1/4 1/4 ρ 1 1,2 ω /2 3/4 1 A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

14 ρ n ω with ρ = {[0, 1/2], [1/2, 3/4], [3/4, 1]} ω 1/2 1/4 1/4 ρ 1 1,2 ω /2 3/4 1 1/4 1/8 1/8 1/4 1/4 ρ 2 1,2 ω /4 3/8 1/2 3/4 1 A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

15 ρ n ω with ρ = {[0, 1/2], [1/2, 3/4], [3/4, 1]} ω 1/2 1/4 1/4 ρ 1 1,2 ω /2 3/4 1 1/4 1/8 1/8 1/4 1/4 ρ 2 1,2 ω /4 3/8 1/2 3/4 1 1/8 1/16 1/16 1/8 1/8 1/8 1/16 1/16 1/8 1/16 1/16 ρ 3 1,2 ω /8 3/16 1/4 3/8 1/2 5/8 11/16 3/4 7/8 15/16 1 A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

16 Definition (Discrepancy) If {π n } is a sequence of partitions, each defined by the points {0 = t0 n < tn 1 <... tn k(n) = 1}, its discrepancy is defined as k(n) D(π n ) = sup 1 χ 0 a<b 1 k(n) [a,b] (ti n ) (b a). i=1 A sequence of partitions {π n } is u.d. if and only if lim D(π n) = 0. n and it is said to have low discrepancy if and only if ( ) 1 D(π n ) = O. k(n) A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

17 M. Drmota & M. Infusino (2012) provided estimates for discrepancy of the Kakutani s sequence of partitions. They are particularly efficient when is rational. log α log(1 α) They also gave estimates for the discrepancy of ρ n ω. The results are significant specially in the case when the lengths of the intervals of ρ, l 1, l 2,..., l m, are rationally related, i.e. the ratios are all rational numbers. log l i log l j A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

18 U.d. sequences of partitions and of points When we want to evaluate an integral, sequences of partitions have two drawbacks, when compared to sequences of points. The first shows up when π n and π n+k do not have significantly many points in common. If, in estimating 1 0 f (t) dt with the average of the values of f in the points determining π n, it is necessary to increase the number of points, one has to recalculate the f in the points of π n+k losing all the work already done. A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

19 We do not face this problem when, for any n, π n+1 is a refinement of π n, as in the case of ρ-refinements. But another problem remains: when calculating the values of f we are not allowed to use the number of points we may consider convenient. This is illustrated in an exemplary way by the sequence of dyadic partitions. Passing from n to n + 1 we have to double the number of points But we know, in this specific case, that there is an elegant way of getting around the problem: we can order in an appropriate way the points of the partitions to obtain the van der Corput sequence of points. A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

20 Definition (Sequential ordering) If {π n } is a sequence of partitions, each defined by the points {0 = t0 n < tn 1 <... tn k(n) = 1}, we say that an ordering of the points tn i is sequential if we first reorder all the points defining π 1, then those defining π 2, and so on. The following result provides a connection between uniformly distributed sequences of partitions and uniformly distributed sequences of points. Theorem (A.V. 2011) A random sequential ordering of the points of a u.d. sequence of partitions {π n } is, with probability 1, a u.d. sequence of points. A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

21 LS-sequences Definition (LS-sequences of partitions) Let L and S be two natural numbers with L 1, S 0 and L + S 2. Let ρ LS be the partition having L long" intervals of length γ and S short" intervals of length γ 2, where γ is the positive (characteristic) root of the characteristic polynomial Sγ 2 + Lγ 1. The LS-sequence of partitions is defined as {ρ n LS ω}. Special case: If S = 0 then L has to be at least 2 and the order of the characteristic polynomial is 1. In this case we get the L-adic partitions, i.e. the partitions of [0, 1] into L n equal parts. A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

22 Theorem (I. Carbone, 2011) The sequence of partitions {ρ n LSω} has low discrepancy if and only if L S. If L = S + 1 the discrepancy is of the order log k(n) k(n) and for L < S + 1 the discrepancy is of the order where c = 1 + log Sγ log γ < 1. 1 k(n) c A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

23 From sequences of partitions to sequences of points In the same paper I. Carbone constructed an algorithm which sequentially orders the points defining the partitions {ρ n LS ω}. When S = 0 and L 2, the algorithm produces the van der Corput sequence in base L. The sequences obtained with this algorithm are called the LS-sequences of points. The most interesting aspect is that the algorithm (improved in a paper submitted recently) turns low discrepancy sequences of partitions into low discrepancy sequences of points. Let us recall that this is the case when L S. A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

24 The Kakutani-Fibonacci sequence A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

25 The Kakutani-Fibonacci sequence β 1 A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

26 The Kakutani-Fibonacci sequence β β 2 β 1 A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

27 The Kakutani-Fibonacci sequence β β 2 β β 3 β 2 β β + β 3 1 A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

28 The Kakutani-Fibonacci sequence β β 2 β β 3 β 2 β β + β β 4 β 3 β 2 β 2 + β 4 β β + β 4 β + β 3 1 A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

29 The Kakutani-Fibonacci sequence of points is particularly interesting, since it connects various areas of uniform distribution: * It is the LS-sequence of points with L = S = 1. * It is a sequential reordering of the sequence of Kakutani s α-refinements, corresponding to the inverse of the golden ratio 5 1 γ =. 2 * it coincides with the β-adic generalized van der Corput sequence for β = Φ, the golden ratio. * It is the orbit T n (0) of an appropriate (Kakutani-Fibonacci) ergodic transformation T constructed by the cutting-stacking method. A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

30 γ γ + γ 31 0 γ 1 0 γ 2 γ γ 1 0 γ 3 γ 2 γ 2 γ A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

31 Theorem (I. Carbone, M.R. Iacò, A. V. 2013) The Kakutani-Fibonacci transformation T constructed with the cutting-stacking method described above is ergodic and measure preserving. Moreover the orbit {T n (0)} coincides with the LS-sequence of points with L = S = 1. Theorem (M. Hofer, M.R. Iacò, R. Tichy 2013) The Kakutani-Fibonacci transformation is uniquely ergodic. A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

32 1 T 1 γ γ 2 T 3 T 2 γ 3 γ 4 T 4 0 γ γ 2 γ 2 + γ 4 γ + γ 3 γ + γ 3 + γ 5 Figure: Partial graph of the Kakutani-Fibonacci transformation A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

33 Higher dimension I We extended Kakutani s splitting procedure (and also ρ-refinements) to higher dimension. By I N = [0, 1] N we denote the unit cube of IR N. By a cartesian N-rectangle (or simply a rectangle) contained in I N we always mean a set of the type R = N j=1 [a j, b j ]. A partition of I N is a finite collection of cartesian rectangles {R i, 1 i k}, which cover I N and have disjoint interiors. A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

34 The definition of uniformly distributed sequence of partitions extends naturally to higher dimension. Definition (Uniform distribution of partitions of [0, 1] N ) Given a sequence of partitions {π n } of I N, with π n = {R n i, 1 i k(n)}, we say that it is uniformly distributed, if for any continuous real valued function f on I N, we have lim n k(n) 1 k(n) i=1 where v n i is any point belonging to R n i. f (vi n ) = f (t) dt, I N A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

35 Definition Fix α ]0, 1[. If π = {R i, 1 i k} is any partition of [0, 1] N, its Kakutani α-refinement απ is obtained by splitting all the rectangles of π having maximal N-dimensional measure λ N in two rectangles, dividing in two segments their longest side such that the left and right part have length proportional to α and 1 α, respectively. If the rectangle R i has several sides with the same length, we split the side with the smallest coordinate index j. A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

36 Let us denote again by ω the trivial partition of I N. By α n ω we indicate the n-th α-partition of ω. The following theorem extends Kakutani s theorem to higher dimension. Theorem (I. Carbone, A.V., 2007) The sequence of partitions {α n ω} is uniformly distributed. The construction of {α n ω} and the proof are intrinsically higher-dimensional. A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

37 Kakutani s partition of [0, 1] 2 for α = 2/3 1" 0" " " """ 1" A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

38 Kakutani s partition of [0, 1] 2 for α = 2/3 1" 0" " " """ 1" A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

39 Kakutani s partition of [0, 1] 2 for α = 2/3 1" 0" " " """ 1" A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

40 Kakutani s partition of [0, 1] 2 for α = 2/3 1" 0" " " """ 1" A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

41 Higher dimension II If {x n } and {y n } are an (L 1, S 1 )- and (L 2, S 2 )-sequence, respectively, then it is natural to ask how does the Halton LS-sequence" {(x n, y n )} behave in [0, 1] 2. Of course the question becomes even more interesting in higher dimensions. We made some experiments which gave interesting results. A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

42 points L=1, S= L=3, S=1 A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

43 points L=4, S= L=S=1 A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

44 It is not known when are the Halton-LS sequences u.d. There are just two nice and interesting negative results due to C. Aistleitner, M. Hofer and V. Ziegler. Assume in both theorems that {x n } and {y n } are an (L 1, S 1 )- and (L 2, S 2 )-sequence, respectively. Theorem 1 If L i S i for i = 1, 2 and there exist integers r and s such that γr 1 γ2 s then {(x n, y n )} is not dense in [0, 1] 2. Q, Theorem 2 If gcd(l 1, S 1, L 2, S 2 ) > 1 then {(x n, y n )} is not dense in [0, 1] 2. A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30

45 Thank you A. Volčič (Università della Calabria) U.d. sequences of partitions Linz, October 14-18, / 30