A von Neumann theorem for uniformly distributed sequences of partitions

A v on Neumann theorem for uniformly distributed sequences of partitions Ingrid Carb one and Aljo ˇ sa V ol ˇ ci ˇ c Universit` a della Calabr ia Abstract In this pap er w e consider p ermuta tions of sequences of partitions, obtaining a result wh ic h parallels v on Neumann’s theorem on p er- m utations of dense sequ ences and uniformly distributed sequences o f p oint s. 1 In tro ductio n S. Kakutani s tudied in [K] the not ion of uniformly d i s tribute d se quenc es o f p artitions of the in terv al [0 , 1]. He in tro duced the following construction. Fix a n um b er α ∈ ]0 , 1[. If π is an y partition of [0 , 1 ], its α -refinemen t, denoted b y α π , is o btained sub dividing the longest in terv al(s) of π in prop or t io n α/ (1 − α ). By α n π we de not e the α -refinemen t of α n − 1 π . Let ω = { [0 , 1] } b e the trivial pa rtition of [0 , 1]. The sequence { α n ω } will b e called t he Kakutani α -sequence. Definition 1.1 Giv en a s equence of partitions { π n } of [0 , 1], with π n = { [ t n i − 1 , t n i ] , 1 ≤ i ≤ k ( n ) } , w e sa y that it is uniformly d istribute d (u.d.) if for an y con tin uous function f on [0 , 1] w e hav e lim n →∞ 1 k ( n ) k ( n ) X i =1 f ( t n i ) = Z 1 0 f ( t ) d t (1) or, eq uiv alen tly , if the sequence of discrete measure s 1 k ( n ) k ( n ) X i =1 δ t n i (2) 1 con v erges we akly to the Leb esgue measure λ restricted to [0 , 1]. Here δ t is the Dirac measure concen trated in t . W e can no w state Ka kutani’s result ( see [K] for more details). Theorem 1.2 The se quenc e { α n ω } i s uniformly distribute d. This was a pa r tial answ er to the fo llo wing question p osed b y t he ph ysicist H. Araki, whic h regarded r andom splittings of the in terv al [0 , 1]. Let X 1 b e c ho osen ra ndomly (with respect to uniform distribution) on [0 , 1] and then, inductiv ely , once X 1 , X 2 , . . . , X n ha v e b een c hosen, let X n +1 b e a p oin t c hosen randomly (a nd uniformly) in the largest of the n + 1 in terv als determined b y the previous n p oints. Can w e conclude that the asso ciated sequence of cum ulative distribution functions con v erges uniformly , with proba bility 1, to the distribution function of the uniform ra ndo m v ariable o n [0 , 1]? By a theorem due to P olya, this is equiv alent to the condition expres sed b y (1). This ques tio n has been studied in [vZ] and [L] and later on in [PvZ]. Note that in the probabilistic setting w e ma y neglect the p o ssibilit y that the pa r tition obtained in the n - th step ha s more than one in terv al of maximal length, since t his ev en t has probabilit y zero. This is not the case for Kaku- tani’s construction, since for ev ery α the partition { α n ω } ha s, for infinitely man y v alues of n , more than one in terv al of maximal length. In [ChV] the not ion of (deterministic) u.d. sequence s of partitions has b een extended to probabilit y spaces on complete metric spaces. In [CV] Kaktuani’s splitting procedure has b een ex tended to higher dimension. Recen tly some new results re viv ed the in terest for the sub ject. In [V] we in tro duced the concept of ρ -r efinemen t of a partition π , whic h generalizes Kakutani α - sequence, and w e pro v ed that the seq uence { ρ n ω } is also u.d. W e a lso inv estigated in [V] the connections of the theory of u.d. se- quences of partitions to the w ell-established theory of uniformly distributed (u.d.) sequences of p oints , sho wing how is it p ossible to asso ciate (many) u.d. seque nces of p oints to an y u.d. seque nce of partitions. A sequence of p oints { x n } is u.d. on [0 , 1] if for ev ery con tinuous function f on [0 , 1] w e hav e lim n →∞ 1 n n X i =1 f ( x i ) = Z 1 0 f ( t ) d t . Since our metho ds allow to construct new u.d. sequence s of p oints, w e think these connections a r e in teresting in view of p ossible a pplications to the 2 quasi-Mon te Carlo metho d, whic h is in the last decades the main motiv ation for the study of u.d. sequences of p oin ts (see [N]). W e contin ue to compare t he tw o theories in the presen t pap er, where w e are concerned with the pro p ert y analog o us to the one studied b y v on Neumann in [vN], where it is prov ed the follo wing theorem. Theorem 1.3 If { x n } is a dense s e quenc e o f p oints in [0 , 1] , then ther e exists a r e arr angement of these p oints, { x n k } , whic h is uniformly d istribute d. As we shall see, there is a remark able differenc e b et wee n von Neumann’s result and its extensions to Borel measures on [0 , 1] and the corresp onding result w e obtain for sequenc es of partitions (Theorem 2 .2 ). W e shall commen t on that in the last s ection of this pap er. One of the consequences of v on Nemann’s result is that there a re man y u.d. seque nces of p oints. Our purp ose is analogo us: w e w an t to sho w that there are many u.d. sequence s o f partitions. Before we pro ceed, w e need to define the concepts of p erm utat io n of a partition and that of densit y of a sequence of partitions. Definition 1.4 Giv en a partit io n π = { [ t i − 1 , t i ] , 1 ≤ i ≤ k } , we denote by l i = t i − t i − 1 the lengh t o f its i -th interv al. The diameter of π , denoted b y diam π , is equal to max 1 ≤ i ≤ k l i . Definition 1.5 Giv en a s equence of partitions { π n } , w e say that it is dense if lim n →∞ diam π n = 0. Definition 1.6 If π = { [ t i − 1 , t i ] , 1 ≤ i ≤ k } is a partition, its p erm utation is a partition π ′ = { [ s h − 1 , s h ] , 1 ≤ h ≤ k } defined by the p oints s h = P h j = 0 l i j , for 0 ≤ h ≤ k , where l 0 = 0 and { i j } , for 1 ≤ i ≤ k , is a p erm utat ion of the indices { 1 , 2 , . . . , k } . W e will denote b y π ! the set of all the permutations of π . Of course, if π splits [0 , 1] in k pa r t s, π has at most k ! permutations: if π has t w o or more in terv a ls of the same length, the n umber of perm utatio ns is smaller than k ! . The extreme case is w hen all the in terv als of π ha v e the same lengh t. Then π ! coincides with the singleton { π } . In the sequel w e shall use the w ell kno wn fact that uniform distribution of par t it ions can b e tested b y means of suitable families of functions and 3 sets (see, for instance, [KN], Section 1 of Chapter 1 and for more general probabilit y spaces, [DT], Sec tion 2 .1). Definition 1.7 W e sa y that a fa mily F of fuctions defin ed on [0 , 1] is deter- mining if a sequence of partitions { π n } is uniformly distributed if a nd only if (1) holds restricted to all f ∈ F . It is v ery simple to sho w that the f amily of all c haracteristic functions of dy adic in terv als I s h = [ h − 1 2 s , h 2 s ], for s ∈ I N and 1 ≤ h ≤ 2 s , is a determining family , i.e. a sequence of partitions { π n } is u.d. if and only if lim n →∞ 1 k ( n ) k ( n ) X i =1 χ I s h ( t n i ) = 1 2 s for all s ∈ I N and 1 ≤ h ≤ 2 s . T o simplify notation, w e shall use in the s equel the followin g notatio n: π n ( I s h ) = 1 k ( n ) k ( n ) X i =1 χ I s h ( t n i ) , (3) denoting the measure defined in (2) by the same sym b ol π n whic h is used for the corresponding partition. 2 The main res ult W e b egin this section by showin g that densit y is a necessary condition for the uniform dis tributio n of a sequence of partitions. Prop osition 2.1 Any uniformly distribute d s e quenc e of p artitions { π n } is dense. Pr o of. Supp ose the contrary . Then there exists an ε > 0 suc h that , for infinitely many indices, diam π n ≥ ε . Denote b y n k these indices and select for eac h n k an in terv al I n k b elonging to π n k ha ving length at least ε . Let m b e a p ositive in teger suc h that 1 m is smaller than ε and consider the p oints x i = i 2 m , for 0 ≤ i ≤ 2 m . Then eac h I n k con tains at least t w o of them a nd t herefore it contains at least one of the interv a ls J i = [ x i − 1 , x i ], for 1 ≤ i ≤ 2 m . It follows that at least one of these in terv als J i (denote it b y 4 J ) is con t ained in infinitely man y I n k ’s. Let no w f b e a con tinuous function ha ving non v anishing in tegra l, whose supp ort is contained in J . Then the sequence A n = 1 k ( n ) k ( n ) X i =1 f ( t n i ) do es not conv erge to R 1 0 f ( t ) dt when n tends to infinit y , since A n k = 0 for all k . This contradicts the uniform distribution of { π n } . ⋆ The densit y is necessary for uniform distribution, but it is of course not sufficien t. Ho w ev er, we ha v e the following result, whic h is t he main result of this paper. Theorem 2.2 If { π n } is a dense se quenc e of p artitions, then ther e exists a se quenc e o f p artitions { σ n } , with σ n ∈ π n ! , which is uniformly distribute d. Pr o of. Let { π n } b e a dense sequenc e of partitions, and let k ( n ) denote the n um b er of in terv als of π n . Since the binary interv als I s h = [ h − 1 2 s , h 2 s ] are a determining fa mily , the conclusion will be a c hea v ed constructing, for eac h n , a p ermutation σ n of π n suc h that lim n →∞ σ n ( I s h ) = 1 2 s for an y s ∈ I N and any 1 ≤ h ≤ 2 s (see Definition 1.7 and form ula (3)). F or any s ∈ I N , there exists n s ∈ I N suc h that diam π n ≤ 1 4 s for a ll n ≥ n s . Of course, if k ( n ) is the num b er of in terv als of π n , w e ha v e that k ( n ) ≥ 4 s for a ll n ≥ n s . W e ma y select a subse quence { n s } so that n s +1 > n s for a ll s ∈ I N . When n < n 1 , we just take σ n = π n . Supp ose now s > 1 a nd let us construct, for eac h n suc h that n s ≤ n < n s +1 , a p erm utat io n σ n of π n suc h that σ n ( I t h ) is close to 1 2 t for all t ≤ s , in a sense whic h will b e made precis e later. First, order t he interv als of the partitio n π n with resp ect to their increas- ing length. Then we shift to the right the first interv al of π n so that its right endp oin t is 1 and w e mov e corresp ondingly the blo c k of a ll the other in terv als to the left. In this wa y the n um b er of the in terv als con tained in I 1 1 remains 5 unc hanged or decreases by one unit. W e rep eat this pro cedure until w e obtain a perm utatio n of π n , denoted by π 1 n , suc h that k ( n ) 2 − 1 ≤ k ( n ) π 1 n ( I 1 h ) ≤ k ( n ) 2 + 1 , 1 ≤ h ≤ 2 . (4) No w w e consider t = 2 and the corresp onding dy adic in terv als I 2 h , with 1 ≤ h ≤ 4. Let us denote b y J 1 1 = h 1 2 − δ 1 1 ( n ) , 1 2 + ˜ δ 1 1 ( n ) h the in terv al of π 1 n con taining 1 2 . Of c ourse, ˜ δ 1 1 ( n ) > 0 and δ 1 1 ( n ) + ˜ δ 1 1 ( n ) ≤ 1 4 s . W e rep eat in I 1 1 \ J 1 1 and in I 1 2 \ J 1 1 the pro cedure used ab ov e in order t o get a con ve nient p ermutation of π 1 n . More precisely , w e kee p J 1 1 fixed and order the tw o collections of in terv als of the partitio n π 1 n con tained in I 1 1 \ J 1 1 and I 1 2 \ J 1 1 resp ectiv ely , with res p ect to their increasing lengh t, and in e ach of them we rep eat the pro cedure described ab ov e, shifting the first in terv al to the righ t a nd the rest of them t o the left and con tinue this pro cedure until the in terv als are distributed, up to o ne unit, propo rtionally to t he lengh ts of I 2 1 and I 2 2 \ J 1 1 , respectiv ely , i.e. pr o p ortionally to α 2 1 ( n ) = 1 4 1 2 − δ 1 1 ( n ) in I 2 1 and α 2 2 ( n ) = 1 4 − δ 1 1 ( n ) 1 2 − δ 1 1 ( n ) in I 2 2 \ J 1 1 . In the same wa y we reorder all the in terv als of π 1 n con tained in I 1 2 \ J 1 1 un til they are distributed prop o rtionally (up to one unit) to α 2 3 ( n ) = 1 4 − ˜ δ 1 1 ( n ) 1 2 − ˜ δ 1 1 ( n ) in I 2 3 \ J 1 1 and α 2 4 ( n ) = 1 4 1 2 − ˜ δ 1 1 ( n ) in I 2 4 . Since δ 1 1 ( n ) + ˜ δ 1 1 ( n ) ≤ 1 4 s , it is clear that all the co efficien ts α 2 h ( n ) may b e estimated in terms of a func tio n of s in the follo wing w a y: 1 4 − 1 4 s 1 2 ≤ α 2 h ( n ) ≤ 1 4 1 2 − 1 4 s , 1 ≤ h ≤ 2 2 . (5) Since the low er and upper estimates in (5) do not dep end on h , and we are assuming that n s ≤ n < n s +1 , in the sequel we will write α 2 s instead of α 2 h ( n ), to simp lify not a tion. W e denote by π 2 n the p erm utation of π 1 n constructed ab o ve (whic h fixes J 1 1 ). Not e that π 2 n do es no t mov e in terv als of π 1 n from I 1 1 to I 1 2 and from I 1 2 to I 1 1 ; therefore for h = 1 , 2 w e can w rite ( k ( n ) π 1 n ( I 1 1 ) − 1) α 2 s − 1 ≤ k ( n ) π 2 n ( I 2 h ) ≤ k ( n ) π 1 n ( I 1 1 )) α 2 s + 1 6 and k ( n ) π 1 n ( I 1 2 ) α 2 s − 1 ≤ k ( n ) π 2 n ( I 2 h ) ≤ k ( n ) π 1 n ( I 1 2 ) α 2 s + 1 for h = 3 , 4. If w e put α 1 s = 1 2 , taking (4 ) in to accoun t w e get k ( n ) α 1 s α 2 s − 2 α 2 s − 1 ≤ k ( n ) π 2 n ( I 2 h ) ≤ k ( n ) α 1 s α 2 s + α 2 s + 1 , 1 ≤ h ≤ 2 2 . Let us make one more step in order to indicate how to pro cede for a n y t ≤ s . Denote by J 2 1 and J 2 2 the tw o in terv als of π 2 n con taining, resp ectiv ely , the p oin ts 1 4 and 3 4 and set J 2 1 =  1 4 − δ 2 1 ( n ) , 1 4 + ˜ δ 2 1 ( n )  and J 2 2 =  3 4 − δ 2 2 ( n ) , 3 4 + ˜ δ 2 2 ( n )  with ˜ δ 2 1 ( n ) , ˜ δ 2 2 ( n ) > 0. W e also note that λ ( J 2 1 ) and λ ( J 2 2 ) are b o th bo unded from abov e b y 1 4 s . Let us pro cedeed as b efore, but now in the four interv als I 2 h \ J 1 1 ∪ J 2 1 ∪ J 2 2 for 1 ≤ h ≤ 2 3 . By reordering all the interv als o f π 2 n con tained in eac h o f them we obtain a new partition π 3 n ∈ π 2 n ! (whic h ke eps fixed J 1 1 , J 2 1 and J 2 2 ) whose in terv als are distributed prop o r t ionally (up to one unit) to certain co efficien ts α 3 h ( n ) (whic h are determined as b efore), all of whic h satisfy the inequalities 1 2 3 − 1 4 s 1 2 2 ≤ α 3 h ( n ) ≤ 1 2 3 1 2 2 − 2 4 s , 1 ≤ h ≤ 2 3 . Here w e note again that, as in fo rm ula (5), the estimates do not dep end on h , and n is fixe d, so in the sequel w e shall write α 3 s instead on α 3 h ( n ). W e note, a s in the previous step, that π 3 n do es not mo ve interv als of π 2 n among the dyadic in terv als I 2 h , with 1 ≤ h ≤ 2 2 . W e easily get k ( n ) α 1 s α 2 s α 3 s − 2 α 2 s α 3 s − 2 α 3 s − 1 ≤ k ( n ) π 3 n ( I 3 h ) ≤ k ( n ) α 1 s α 2 s α 3 s + α 2 s α 3 s + α 3 s + 1 for all 1 ≤ h ≤ 2 3 . 7 If is clear now that the same procedure can b e rep eated for all t ≤ s and that w e can c onstruct partitions π t n ∈ π t − 1 n ! suc h that k ( n ) t Y j = 1 α j s − 2 t X i =2 t Y j = i α j s − 1 ≤ k ( n ) π t n ( I t h ) ≤ k ( n ) t Y j = 1 α j s + t X i =2 t Y j = i α j s + 1 , (6) where α t s , with 2 ≤ t ≤ s , are the prop ortionality co efficien ts whic h satisfy α 1 s = 1 2 and 1 2 t − 1 4 s 1 2 t − 1 ≤ α t s ≤ 1 2 t 1 2 t − 1 − 2 4 s , 2 ≤ t ≤ s. (7) Moreo v er, (6) implies the follo wing estimates k ( n )       π t n ( I t h ) − t Y j = 1 α j s       ≤ 2 t X i =2 t Y j = i α j s + 1 , 1 ≤ h ≤ 2 t , 1 ≤ t ≤ s. (8) No w w e obse rve that the partition π t n do es not mo ve the in terv als of π t − 1 n con tained in eac h dy a dic in terv al I t − 1 h , w ith 1 ≤ h ≤ 2 t − 1 , to another dy adic in terv al I t − 1 k , with 1 ≤ k ≤ 2 t − 1 , i.e. π t − 1 n ( I t − 1 h ) = π t n ( I t − 1 h ) f or a ll t ≤ s , 1 ≤ h ≤ 2 t − 1 . Let us put σ n = π s n and observ e that the condition n s ≤ n < n s +1 implies that n → ∞ if and only if s → ∞ . If w e divide the terms of (8) by k ( n ) and subs titute π t n b y σ n , w e obta in | σ n ( I t h ) − L s ( t ) | ≤ R n,s ( t ) , 1 ≤ h ≤ 2 t , 1 ≤ t ≤ s, (9) where L s ( t ) = t Y j = i α j s , and R n,s ( t ) = 1 k ( n )   2 t X i =2 t Y j = i α j s + 1   . W e will pro vide now estimates for L s ( t ) − 1 2 t and f or R n,s ( t ). T o this purp ose w e observ e that f r om (7) we get for all 1 ≤ t ≤ s 1 2 t t Y j = 2 1 − 2 j 4 s ! ≤ L s ( t ) ≤ 1 2 t 1 Q t j = 2  1 − 2 j 4 s  . 8 If w e substitute all the te rms in the brac k ets of the previous form ula b y the smallest of them, w e get 1 2 t < 1 2 t 1 − 2 t 4 s ! t − 1 ≤ L s ( t ) ≤ 1 2 t 1  1 − 2 t 4 s  t − 1 . W e note that 1 − 2 t 4 s ! 1 − t ≤  4 s 4 s − 2 s  s for all 1 ≤ t ≤ s and elemen tary calculation show s that the sequence of these upp er b ounds, whic h dep end only on s , tends to 1 when s tends to infinit y . Then, f o r all 1 ≤ t ≤ s w e ha v e 0 < L s ( t ) − 1 2 t ≤ 1 2 t  4 s 4 s − 2 s  s − 1 2 t (10) and, consequ ently , L s ( t ) − 1 2 t → 0 when s tends to infinit y . Moreo v er, for R n,s ( t ) w e hav e the follo wing estim at e: 0 < R n,s ( t ) ≤ 2 k ( n ) t X i =2 t Y j = i α j + 1 k ( n ) ≤ 2 s − 2 4 s L s ( s ) + 1 4 s (11) for all 1 ≤ t ≤ s , therefore R n,s ( t ) is b ounded b y a sequence whic h tends to zero when s tends to infinit y . Substituting (10) and (11) in form ula (9), w e get     σ n ( I t h ) − 1 2 t     ≤ R n,s ( t ) + L s ( t ) − 1 2 t ≤ 2 s − 2 4 s L s ( s )+ 1 4 s + 1 2 t  4 s 4 s − 2 s  s − 1  . Putting all things to g ether, w e can conclude therefore that σ n ( I t h ) → 1 2 t when n → ∞ for all t ∈ I N . ⋆ 9 3 Conclus ions W e ha v e cons tructed the u.d. sequence { σ n } fro m a dense se quence { π n } by a precise alg o rithm whic h could b e, in concrete situations, implemen ted step b y step. On the other hand it w ould b e in teresting to hav e b esides our result also a probabilistic stateme nt expre ssed by the following Conjecture I f { π n } is a dense se quenc e of p artitions and if e ach σ n is taken at r andom fr om π n ! for any n ∈ I N , then { σ n } is unifo rm ly dis tribute d with pr ob ability 1 . The ( p ossible) confirmation of the correctness of this conjecture w ould not diminish the v alue of the result presen ted in this pap er. If it is allow ed to compare the questions we are treating with m uc h more imp ortant ones, w e all agree that the Borel theorem on normal num b ers do es not diminish the in terest for fin ding concrete examples of suc h num b ers. It is interesting to note also an essen tial difference b et wee n the theorem due to von Neumann and our result. It is well know n that the c onclusion of Theorem 1.3 holds if the measure λ is substituted by an y Borel pro babilit y on [0 , 1] (see for instance Section 4 of Chapter 2 a nd, f or a more general setting, Section 2 of Chapter 3, of [KN] and the bibliogra ph y cited there). This is no t the case with our result, as it can b e easily seen b y ta king a sequence of partitions { π n } , where each π n splits [0 , 1] in in terv als of equal length. Then for an y n ∈ I N the set of perm utat io ns π n ! is a singleton and λ is the only p ossible limit. This is of course an extreme case and it is not difficult to see by examples that in many cases, if { π n } is a dense sequence of pa rtitions, the corresp ond- ing set M of all the probabilities which are limits of sequence s { σ n } , with σ n ∈ π n !, con tains more than just λ . This observ a t io n rises questions ab out the size of the set M (could it con- tain in some cases all the Borel measures?) and its geometric and to p ological prop erties (is it conv ex, is it closed?). The conjecture and the questions ab out M will b e addressed in subse- quen t pap ers. 10 References [ChV] F. Chersi, A. V ol ˇ ci ˇ c, λ -equidistributed sequences of partit io ns and a theorem o f the de Bruijn-P ost t yp e, Annali Mat. Pur a Appl. (4) 162 (19 92) 23-32. [CV] I. Carb one, A. V olˇ ci ˇ c, Ka kutani splitting pro cedure in hig her dimension, R en d. Ist. Matem. Univ. T rieste 39 (2 0 07) 119-126. [DT] M. Drmota, R. F. Tich y , Se quenc es, dis cr ep ancies and applic ations , Lecture Notes in Mathematics 1651 , Springer V erlag, Berlin, 1997. [K] S. Kakutani, A problem on equidistribution on t he unit in terv al [0 , 1], Me as ur e the o ry ( Pr o c. Conf. , Ob erwolfach, 1975) , pp. 36 9–375. L e ctur e Notes in Math. 541 , Springer, Berlin, 1976. [KN] L. 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