 # A random von Neumann theorem for uniformly distributed sequences of partitions

From the journal Open Mathematics

## Abstract

In this paper, we prove a theorem that confirms, under a supplementary condition, a conjecture concerning random permutations of sequences of partitions of the unit interval.

MSC 2010: 11-xx; 40-xx; 60-xx

## 1 Introduction

The general study of uniformly distributed sequences of partitions was initiated in , inspired by a beautiful construction and result from the study by Kakutani . The subject is closely related to the theory of uniformly distributed sequences, initiated in . There are two classical references for the subject:  and .

Kakutani took the interval I = [ 0 , 1 ] , a number α ] 0 , 1 [ and divided the interval in proportion α : 1 α . 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 [ . 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 { α n I } ) is uniformly distributed, which means that if α k I = { 0 < t 1 k < t 2 k < < t N k n < 1 } is the kth partition, then

lim k 1 N k i = 1 N k f ( t i k ) = 0 1 f ( t ) d t ,

for every continuous function f .

In other words, the discrete measure concentrated in the points t i k converges weakly to the Lebesgue measure on [ 0 , 1 ] .

The construction has been generalized in . Let ρ be any non trivial finite partition of I .

In the first step, the longest interval(s) of ρ is subdivided positively homothetically to ρ . The partition obtained in this manner is denoted by ρ 2 I . In the second step, the same procedure is repeated on the longest interval(s), operating with ρ on ρ 2 I . Iteration of this procedure leads to a sequence of partitions { ρ n I } .

If ρ = { [ 0 , α [ , [ α , 1 [ } , one gets Kakutani’s sequence.

The following theorem includes the results of the study by Kakutani (, Theorem 2.7]).

## Theorem 1

The sequence { ρ n I } 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 -refinements, the so-called L S -sequences. The subject was initiated by the present author in , and it is connected with the van der Corput sequences of points.

L S -sequences are constructed starting from the partition ρ L S made of L + S intervals ( L and S are positive integers) of length β and β 2 , respectively, where β is the positive solution of the equation L β + S β 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 .

## Theorem 2

If { x n } is a dense sequence of points in [ 0 , 1 ] , then there exists a rearrangement of these points, { x n k } , 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.

## Definitions

Given a partition π = { [ t i 1 , t i ] , 1 i N } , we denote by l i = t i t i 1 the length of its ith interval.

The diameter of π , denoted by L , is equal to max 1 i N l i .

Given a sequence of partitions { π k } , we say that it is dense if, denoted by L k the diameter of π k , lim k L k = 0 .

If π = { [ t i 1 , t i ] , 1 i N } is a partition, its random permutation is a partition π = { [ s h 1 , s h ] , 1 h N } defined by the points s h = j = 0 h l i j , for 0 h N , where s 0 = 0 and the indices { i j } , are successively taken at random, with probability 1 N , from the set { i s : 1 s N } .

We will denote by π ! the set of all the N ! permutations of π .

## 2 Main results

In a previous paper , we proved the following result.

## Proposition 3

If { π n } is a dense sequence of partitions, then there exists a sequence of partitions { σ n } , with σ n π n ! , which is uniformly distributed.

In the same paper, we made the following conjecture.

## Conjecture

If { π n } is a dense sequence of partitions and we select at random a partition σ k π n ! , then { σ k } is uniformly distributed with probability 1.

We need some preliminary calculations.

Let q ] 0 , 1 [ and denote by N k ( q ) the integer such that

N k ( q ) N k q < N k ( q ) + 1 N k .

Consider a sequence { q m } of points, which is dense in [ 0 , 1 ] . For later convenience, we will denote by N k ( m ) the integer N k ( q m ) .

Select at random from the N k intervals of π k , with probability 1 N k , N k ( m ) intervals. Denote by ξ i k the length of the interval selected in the ith draw ( 1 i N k ( m ) ) and consider the random variable

η k m = i = 1 N k ( m ) ξ i k .

Obviously,

E ( η k m ) = i = 1 N k ( m ) E ( ξ i k ) = i = 1 N k ( m ) 1 N k = N k ( m ) N k ,

hence,

E ( η k m ) q m 1 N k .

It is easy to see that the second moment of η k m is uniformly bounded for any sequence of partitions.

This, together with the independence of the η k m ’s (for k N ), would allow us to apply the strong law of large numbers and to conclude that, when k tends to infinity, the sequence η k m tends to 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 { π k } .

A simple sufficient assumption is expressed as follows:

i = 1 L k 2 < .

## Theorem 4

If the series of squares of diameters of { π k } is convergent, then its random permutations σ k are uniformly distributed with probability 1.

## Proof

We have

Var ( η k m ) = E i = 1 N k ( m ) ξ i k i = 1 N k ( m ) 1 N k 2 = E i = 1 N k ( m ) ξ i k 1 N k 2 = E i = 1 N k ( m ) ( ξ i k ) 2 2 i = 1 N k ( m ) ξ i k 1 N k + i = 1 N k ( m ) 1 N k 2 = E i = 1 N k ( m ) ( ξ i k ) 2 2 1 N k E i = 1 N k ( m ) ξ i k + E i = 1 N k ( m ) 1 N k 2 E i = 1 N k ( ξ i k ) 2 2 N k E i = 1 N k ξ i k + i = 1 N k 1 N k = E i = 1 N k ( ξ i k ) 2 i = 1 N k 1 N k i = 1 N k L k 2 1 < i = 1 N k L k 2 .

Apply now the Čebišëv inequality. By our assumption, we have, for every ε > 0 (and every m ),

k = 1 P ( η k m E ( η k m ) > ε ) n = 1 Var ( η k m ) ε 2 < .

Recalling that E ( η k m ) tends to q m and applying the Borel-Cantelli lemma [9, Theorem 4.2.1], we obtain that

lim k η k m = q m

almost surely for every m N .

The set { q m } is countable; therefore, the aforementioned limit holds almost surely for all the values of m simultaneously.

Observe now that lim k N k ( q ) N k is an increasing function of q . Therefore, it follows that, almost surely,

lim k N k ( q ) N k = q

for every q [ 0 , 1 ] .

In other words, the empirical distribution function F k of σ k tends almost surely to the distribution function of the random variable U uniformly distributed on [ 0 , 1 ] .

On the other hand, convergence in distribution is known to be equivalent to weak convergence, so the desired conclusion follows.□

## Acknowledgements

The author wishes to thank the referees for their useful comments.

1. Conflict of interest: The author states no conflict of interest.

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