Access the full text.

Sign up today, get DeepDyve free for 14 days.

Open Mathematics
, Volume 20 (1): 4 – Jan 1, 2022

/lp/de-gruyter/a-random-von-neumann-theorem-for-uniformly-distributed-sequences-of-G6maIUhQuS

- Publisher
- de Gruyter
- Copyright
- © 2022 Ingrid Carbone, published by De Gruyter
- ISSN
- 2391-5455
- eISSN
- 2391-5455
- DOI
- 10.1515/math-2022-0016
- Publisher site
- See Article on Publisher Site

1IntroductionThe general study of uniformly distributed sequences of partitions was initiated in [1], inspired by a beautiful construction and result from the study by Kakutani [2]. The subject is closely related to the theory of uniformly distributed sequences, initiated in [3]. There are two classical references for the subject: [4] and [5].Kakutani took the interval I=[0,1]I=\left[0,1], a number α∈]0,1[\alpha \in ]0,1{[}and divided the interval in proportion α:1−α\alpha :1-\alpha . Then, he divided the longest interval of this partition in the same proportion and iterated the procedure dividing always the longest interval of the nth partition, so as to obtain a sequence of partitions of ]0,1[]0,1{[}. If at a certain step there were two or more intervals of maximal length, they were divided simultaneously.Kakutani proved that this sequence of partitions (denote it by {αnI}\left\{{\alpha }^{n}I\right\}) is uniformly distributed, which means that if αkI={0<t1k<t2k<⋯<tNkn<1}{\alpha }^{k}I=\left\{0\lt {t}_{1}^{k}\lt {t}_{2}^{k}\hspace{0.33em}\lt \cdots \lt {t}_{{N}_{k}}^{n}\lt 1\right\}is the kth partition, then limk→∞1Nk∑i=1Nkf(tik)=∫01f(t)dt,\mathop{\mathrm{lim}}\limits_{k\to \infty }\frac{1}{{N}_{k}}\mathop{\sum }\limits_{i=1}^{{N}_{k}}f\left({t}_{i}^{k})=\underset{0}{\overset{1}{\int }}f\left(t){\rm{d}}t,for every continuous function ff.In other words, the discrete measure concentrated in the points tik{t}_{i}^{k}converges weakly to the Lebesgue measure on [0,1]\left[0,1].The construction has been generalized in [1]. Let ρ\rho be any non trivial finite partition of II.In the first step, the longest interval(s) of ρ\rho is subdivided positively homothetically to ρ\rho . The partition obtained in this manner is denoted by ρ2I{\rho }^{2}I. In the second step, the same procedure is repeated on the longest interval(s), operating with ρ\rho on ρ2I{\rho }^{2}I. Iteration of this procedure leads to a sequence of partitions {ρnI}\left\{{\rho }^{n}I\right\}.If ρ={[0,α[,[α,1[}\rho =\left\{{[}0,\alpha {[},{[}\alpha ,1{[}\right\}, one gets Kakutani’s sequence.The following theorem includes the results of the study by Kakutani ([1], Theorem 2.7]).Theorem 1The sequence {ρnI}\left\{{\rho }^{n}I\right\}is uniformly distributed.There are interesting connections between the theory of u.d. sequences of partitions and u.d. sequences of points. This connection is far reaching in the construction of a significant subclass of ρ\rho -refinements, the so-called LSLS-sequences. The subject was initiated by the present author in [6], and it is connected with the van der Corput sequences of points.LSLS-sequences are constructed starting from the partition ρLS{\rho }_{LS}made of L+SL+Sintervals (LLand SSare positive integers) of length β\beta and β2{\beta }^{2}, respectively, where β\beta is the positive solution of the equation Lβ+Sβ2=1L\beta +S{\beta }^{2}=1.The present paper is concerned with a result in the domain of uniformly distributed sequences of partitions, related to a proposition by von Neumann for uniformly sequences of points [7].Theorem 2If {xn}\left\{{x}_{n}\right\}is a dense sequence of points in [0,1]\left[0,1], then there exists a rearrangement of these points, {xnk}\left\{{x}_{{n}_{k}}\right\}, which is uniformly distributed.One of the consequences of von Neumann’s result is that there are many u.d. sequences of points.We will now introduce the definitions we need.DefinitionsGiven a partition π={[ti−1,ti],1≤i≤N}\pi =\left\{\left[{t}_{i-1},{t}_{i}],1\le i\le N\right\}, we denote by li=ti−ti−1{l}_{i}={t}_{i}-{t}_{i-1}the length of its ith interval.The diameter of π\pi , denoted by LL, is equal to max1≤i≤Nli{\max }_{1\le i\le N}{l}_{i}.Given a sequence of partitions {πk}\left\{{\pi }_{k}\right\}, we say that it is dense if, denoted by Lk{L}_{k}the diameter of πk{\pi }_{k}, limk→∞Lk=0{\mathrm{lim}}_{k\to \infty }{L}_{k}=0.If π={[ti−1,ti],1≤i≤N}\pi =\left\{\left[{t}_{i-1},{t}_{i}],1\le i\le N\right\}is a partition, its random permutation is a partition π′={[sh−1,sh],1≤h≤N}\pi ^{\prime} =\left\{\left[{s}_{h-1},{s}_{h}],1\le h\le N\right\}defined by the points sh=∑j=0hlij,{s}_{h}={\sum }_{j=0}^{h}{l}_{{i}_{j}},for 0≤h≤N0\le h\le N, where s0=0{s}_{0}=0and the indices {ij}\left\{{i}_{j}\right\}, are successively taken at random, with probability 1N\frac{1}{N}, from the set {is:1≤s≤N}\left\{{i}_{s}:1\le s\le N\right\}.We will denote by π!\pi \hspace{0.1em}\text{!}\hspace{0.1em}the set of all the N!N\!permutations of π\pi .2Main resultsIn a previous paper [8], we proved the following result.Proposition 3If {πn}\left\{{\pi }_{n}\right\}is a dense sequence of partitions, then there exists a sequence of partitions {σn}\left\{{\sigma }_{n}\right\}, with σn∈πn!{\sigma }_{n}\in {\pi }_{n}\hspace{0.1em}\text{!}\hspace{0.1em}, which is uniformly distributed.In the same paper, we made the following conjecture.ConjectureIf {πn}\left\{{\pi }_{n}\right\}is a dense sequence of partitions and we select at random a partition σk∈πn!{\sigma }_{k}\in {\pi }_{n}\hspace{0.1em}\text{!}\hspace{0.1em}, then {σk}\left\{{\sigma }_{k}\right\}is uniformly distributed with probability 1.We need some preliminary calculations.Let q∈]0,1[q\in ]0,1{[}and denote by Nk(q){N}_{k}\left(q)the integer such that Nk(q)Nk≤q<Nk(q)+1Nk.\frac{{N}_{k}\left(q)}{{N}_{k}}\le q\lt \frac{{N}_{k}\left(q)+1}{{N}_{k}}.Consider a sequence {qm}\left\{{q}_{m}\right\}of points, which is dense in [0,1]\left[0,1]. For later convenience, we will denote by Nk(m){N}_{k}\left(m)the integer Nk(qm){N}_{k}\left({q}_{m}).Select at random from the Nk{N}_{k}intervals of πk{\pi }_{k}, with probability 1Nk\frac{1}{{N}_{k}}, Nk(m){N}_{k}\left(m)intervals. Denote by ξik{\xi }_{i}^{k}the length of the interval selected in the ith draw (1≤i≤Nk(m)1\le i\le {N}_{k}\left(m)) and consider the random variable ηkm=∑i=1Nk(m)ξik.{\eta }_{k}^{m}=\mathop{\sum }\limits_{i=1}^{{N}_{k}\left(m)}{\xi }_{i}^{k}.Obviously, E(ηkm)=∑i=1Nk(m)E(ξik)=∑i=1Nk(m)1Nk=Nk(m)Nk,E\left({\eta }_{k}^{m})=\mathop{\sum }\limits_{i=1}^{{N}_{k}\left(m)}E\left({\xi }_{i}^{k})=\mathop{\sum }\limits_{i=1}^{{N}_{k}\left(m)}\frac{1}{{N}_{k}}=\frac{{N}_{k}\left(m)}{{N}_{k}},hence, ∣E(ηkm)−qm∣≤1Nk.| E\left({\eta }_{k}^{m})-{q}_{m}| \le \frac{1}{{N}_{k}}.It is easy to see that the second moment of ηkm{\eta }_{k}^{m}is uniformly bounded for any sequence of partitions.This, together with the independence of the ηkm{\eta }_{k}^{m}’s (for k∈Nk\in {\mathbb{N}}), would allow us to apply the strong law of large numbers and to conclude that, when kktends to infinity, the sequence ηkm{\eta }_{k}^{m}tends to qm{q}_{m}in the Cesàro mean (and nothing more, at least following this line of thought).But this is not what we were looking for.If we want to identify a class for which the conjecture is true, we have to make some assumptions on the sequence {πk}\left\{{\pi }_{k}\right\}.A simple sufficient assumption is expressed as follows: ∑i=1∞Lk2<∞.\mathop{\sum }\limits_{i=1}^{\infty }{L}_{k}^{2}\lt \infty .Theorem 4If the series of squares of diameters of {πk}\left\{{\pi }_{k}\right\}is convergent, then its random permutations σk{\sigma }_{k}are uniformly distributed with probability 1.ProofWe have Var(ηkm)=E∑i=1Nk(m)ξik−∑i=1Nk(m)1Nk2=E∑i=1Nk(m)ξik−1Nk2=E∑i=1Nk(m)(ξik)2−2∑i=1Nk(m)ξik1Nk+∑i=1Nk(m)1Nk2=E∑i=1Nk(m)(ξik)2−21NkE∑i=1Nk(m)ξik+E∑i=1Nk(m)1Nk2≤E∑i=1Nk(ξik)2−2NkE∑i=1Nkξik+∑i=1Nk1Nk=E∑i=1Nk(ξik)2−∑i=1Nk1Nk≤∑i=1NkLk2−1<∑i=1NkLk2.\begin{array}{rcl}\hspace{0.1em}\text{Var}\hspace{0.1em}\left({\eta }_{k}^{m})& =& E\left({\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\xi }_{i}^{k}-\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}\frac{1}{{N}_{k}}\right)}^{2}\right)\\ & =& E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\left({\xi }_{i}^{k}-\frac{1}{{N}_{k}}\right)}^{2}\right)\\ & =& E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\left({\xi }_{i}^{k})}^{2}-2\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\xi }_{i}^{k}\frac{1}{{N}_{k}}+\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}\frac{1}{{N}_{k}^{2}}\right)\\ & =& E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\left({\xi }_{i}^{k})}^{2}\right)-2\frac{1}{{N}_{k}}E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\xi }_{i}^{k}\right)+E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}\frac{1}{{N}_{k}^{2}}\right)\\ & \le & E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}{\left({\xi }_{i}^{k})}^{2}\right)-\frac{2}{{N}_{k}}E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}{\xi }_{i}^{k}\right)+\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}\frac{1}{{N}_{k}}\\ & =& E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}{\left({\xi }_{i}^{k})}^{2}\right)-\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}\frac{1}{{N}_{k}}\le \mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}{L}_{k}^{2}-1\lt \mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}{L}_{k}^{2}.\end{array}Apply now the Čebišëv inequality. By our assumption, we have, for every ε>0\varepsilon \gt 0(and every mm), ∑k=1∞P(∣ηkm−E(ηkm)∣>ε)≤∑n=1∞Var(ηkm)ε2<∞.\mathop{\sum }\limits_{k=1}^{\infty }P\left(| {\eta }_{k}^{m}-E\left({\eta }_{k}^{m})| \gt \varepsilon )\le \mathop{\sum }\limits_{n=1}^{\infty }\frac{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({\eta }_{k}^{m})}{{\varepsilon }^{2}}\lt \infty .Recalling that E(ηkm)E\left({\eta }_{k}^{m})tends to qm{q}_{m}and applying the Borel-Cantelli lemma [9, Theorem 4.2.1], we obtain that limk→∞ηkm=qm\mathop{\mathrm{lim}}\limits_{k\to \infty }{\eta }_{k}^{m}={q}_{m}almost surely for every m∈Nm\in {\mathbb{N}}.The set {qm}\left\{{q}_{m}\right\}is countable; therefore, the aforementioned limit holds almost surely for all the values of mmsimultaneously.Observe now that limk→∞Nk(q)Nk{\mathrm{lim}}_{k\to \infty }\frac{{N}_{k}\left(q)}{{N}_{k}}is an increasing function of qq. Therefore, it follows that, almost surely, limk→∞Nk(q)Nk=q\mathop{\mathrm{lim}}\limits_{k\to \infty }\frac{{N}_{k}\left(q)}{{N}_{k}}=qfor every q∈[0,1]q\in \left[0,1].In other words, the empirical distribution function Fk{F}_{k}of σk{\sigma }_{k}tends almost surely to the distribution function of the random variable UUuniformly distributed on [0,1]\left[0,1].On the other hand, convergence in distribution is known to be equivalent to weak convergence, so the desired conclusion follows.□

Open Mathematics – de Gruyter

**Published: ** Jan 1, 2022

**Keywords: **number theory; discrepancy; uniformly distributed sequences of partitions; probability theory; 11-xx; 40-xx; 60-xx

Loading...

You can share this free article with as many people as you like with the url below! We hope you enjoy this feature!

Read and print from thousands of top scholarly journals.

System error. Please try again!

Already have an account? Log in

Bookmark this article. You can see your Bookmarks on your DeepDyve Library.

To save an article, **log in** first, or **sign up** for a DeepDyve account if you don’t already have one.

Copy and paste the desired citation format or use the link below to download a file formatted for EndNote

Access the full text.

Sign up today, get DeepDyve free for 14 days.

All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.