Low discrepancy sequences failing Poissonian pair correlations

03/05/2019 ∙ by Verónica Becher, et al. ∙ University of Buenos Aires 0

M. Levin defined a real number x that satisfies that the sequence of the fractional parts of (2^n x)_n≥ 1 are such that the first N terms have discrepancy O(( N)^2/ N), which is the smallest discrepancy known for this kind of parametric sequences. In this work we show that the fractional parts of the sequence (2^n x)_n≥ 1 fail to have Poissonian pair correlations. Moreover, we show that all the real numbers x that are variants of Levin's number using Pascal triangle matrices are such that the fractional parts of the sequence (2^n x)_n≥ 1 fail to have Poissonian pair correlations.

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1. Introduction and statement of results

A sequence of real numbers in the unit interval is said to have Poissonian pair correlations if for all non-negative real numbers ,

where

and is the distance between and its nearest integer. The function counts the number of pairs for , , of points which are within distance at most of each other, in the sense of distance on the torus. If for all

then the asymptotic distribution of the pair correlations of the sequence is Poissonian, and this explains that the property is referred as having Poissonian pair correlations. Almost surely a sequence of independent identically distributed random variables in the unit interval has this property. Several particular sequences have been proved to have the property, for example

 [4]. It is known that for almost all real numbers  has the property when is integer valued and is lacunary [11]; also when (or a higher polynomial) [14, 10]. However, for specific values such as and it is not known whether has Poissonian pair correlations or not.

The property of Poissonian pair correlations implies uniform distribution modulo 

, this was only recently proved in [1, Theorem 1] and also in [5, Corollary 1.2]. The converse does not always hold. Several uniformly distributed sequences of the form where is an integer greater than  and is a constant were proved to fail the property of Poissonian pair correlations. Pirsic and Stockinger [9] proved it for Champernowne’s constant (defined in base ). Larcher and Stockinger [6] proved it for a Stoneham number [12] and for every real number having an expansion which is an infinite de Bruijn word (see [3, 13] for the presentation of these infinite words). Larcher and Stockinger in [7] also show the failure of the property for other sequences of the form .

In this paper we show that the sequence , where is the real number defined by Levin in [8, Theorem 2], fails to have Poissonian pair correlations. Levin’s number is defined constructively using Pascal triangle matrices and satisfies that the discrepancy of the first terms of the sequence is . This is the smallest discrepancy bound known for sequences of the form for some real number .

We also show that each of the real numbers considered by Becher and Carton in [2] are such that the sequence fails to have Poissonian pair correlations. These numbers are variants of Levin’s number because they are defined using rotations of Pascal triangle matrices and the sequence has the same low discrepancy as that obtained by Levin.

1.1. Levin’s number

We start by defining the number given by Levin in [8, Theorem 2] and further examined in [2]. As usual, we write

to denote the field of two elements. In this work, we freely make the identification between binary words and vectors on

. We define recursively a sequence of matrices on :

The first elements of this sequence, for example, are:

Let be a non-negative integer and let . The matrix is upper triangular with s on the diagonal, hence it is non-singular. Then, if

is the enumeration of all vectors of length  in lexicographical order, the sequence

ranges over all vectors of length . We obtain the -th block of  by concatenation of the terms of that sequence:

Levin’s constant is defined as the infinite concatenation

The expansion of in base starts as follows (the spaces are just for convenience):

Now we introduce a family of constants which have similar properties to those of . Let be the rotation that takes a word and moves its last letter at the beginning: that is, . We are going to use to define a family of matrices obtained by selectively rotating some of the columns of .

As before, assume is a non-negative integer and let . We say that a tuple of non-negative integers is suitable if

Let denote the columns of . Then, define

For example, by taking we have different possible matrices, one for every choice of :

As before, we let and be the increasingly ordered sequence of all vectors in . We say that a word is an -affine necklace if it can be written as the concatenation for some and a suitable tuple  with and for .

Finally, we define to be the set of all binary words that can be written as an infinite concatenation where every is a -affine necklace. Note that , by taking everywhere.

The rest of this note is devoted to proving the following result:

Theorem 1.

For all , the sequence of fractional parts of does not have Poissonian pair correlations.

2. Lemmas

Now we prove some necessary results. We present in an alternating manner results about and its corresponding generalizations to the family of matrices .

Lemma 1.

For all , is triangular and all entries in its diagonal are ones. In particular, is non singular.

Proof.

This is easily proven with induction. satisfies the lemma, and if satisfies it, then satisfies it too. ∎

Lemma 2.

For all non-negative and for every suitable tuple , is non-singular.

Proof.

This fact is proven in [2, Lemma 4]. ∎

Lemma 3.

For all and for all even , and are complementary vectors. That is, the -th coordinate of equals zero if and only if the -th coordinate of equals one.

Proof.

The sequence is lexicographically ordered and hence the last entry of is zero whenever is even. Therefore, only differs from in the last entry

To simplify notation, from now on we write  for the complementary vector of . Note that Lemma 3 implies that can be written as a concatenation of words of the form .

Lemma 4.

For all non-negative , for all even , and for every suitable tuple , the vectors and are complementary.

Proof.

The last coordinate of is zero by definition. Therefore, the last column of is the same as the last column of ; that is, it is the vector of ones. The same argument used to prove Lemma 3 applies. ∎

We say that a vector is even if its last entry is . Hence, when is even, is an even vector.

Lemma 5.

Let be a non-negative integer and . The subspace of all even vectors of length ,

is invariant under . Furthermore, is an even vector if and only if is an even vector.

Proof.

By Lemma 1,

is upper triangular and its diagonal is comprised by ones. This implies that all of its columns except the last are even vectors. Therefore, the only way to obtain an odd vector via the computation

is that itself is odd. ∎

Lemma 6.

Let be a non-negative integer and . Depending on , there are two distinct possibilities:

  1. The subspace of even vectors is invariant under . In this case, is an even vector if and only if is an even vector.

  2. The subspace is in bijection with the subspace via . In this case, is an even vector if and only if has a zero in its first coordinate.

Proof.

Take such that and for all . Combining for and gives that is either zero or one. We consider both possibilities separately.

First, suppose that equals one. Then, all entries of before it must be greater than one. That means that, when building from , all of its columns except the last are rotated at least one position. Fix an index such that , and consider the -th column of the matrix . We show that the first element of is zero.

By Lemma 1, we know that is triangular. That means that the elements are necessarily zeros. But the first element of is ; and from the inequality it follows that is greater or equal than . Therefore, the first element of is zero.

Because is any index between and , it follows that the first columns of have a zero as their first coordinate. If is an even vector, is a linear combination of vectors which start with zero, and therefore also starts with a zero. Conversely, if begins with a one then must be an odd vector, because it’s a linear combination of elements that start with a zero and the last column of , which is the vector of ones. We conclude that is an even vector if and only if starts with a zero.

The case where equals zero is analogous, and we give an outline of the proof. First, prove that the first columns of are even vectors. Then, is even if and only if is even. ∎

3. Proof of the main theorem

We first prove the theorem for and at the end we explain how to generalize the result to each number in the family . For any given non-negative , we set and show that for an appropriate choice of increasing which depends on and , diverges. For this we show that some selected patterns have too many occurrences in . More precisely, we count occurrences of binary words of length ,

such that

The reason for this choice comes from Lemma 3 and it will soon become clear. We need some terminology. Given a word as above and an occurrence of in ,

  1. let be the number of zeros in ;

  2. let be the index such that the occurs in ;

  3. let be the position in  that matches with the -th (that is the last) symbol of .

Figure 1. An occurrence of

We require to be an even number and to be in the range . The latter is to prevent the word from spanning over more than two words, and the former is to ensure that a match for the first letters automatically yields a match for the last letters (a combination of Lemma 3 and the hypothesis over ). In addition, by Lemma 5 we know that is an even word, and therefore must be zero.

We fix and count all possible occurrences in of every possible word . There are exactly

words with zeros in . For every one of them, we have a choice of different , because we know that must be zero. We claim that each of those choices for correspond to an actual occurrence of in . Let’s suppose that the binary word , whose length is , starts in and continues in . Then, it must hold that, for some ,

and

So, Lemma 3 allows us to conclude that

Figure 2. Given an and a choice of , it is possible to find an unique position within where the word occurs with alignment .

By Lemma 1, we know that there exists some that satisfies this equation and by Lemma 5, must be an even number. Therefore, given a choice of there is an occurrence of . We conclude that for every choice of , and for every word with zeros in , we have exactly occurrences within .

We now prove that the sequence of fractional parts of does not have Poissonian pair correlations. Take and . We prove that . In order to do that, we note that two different occurrences of the same word correspond to two different suffixes of that share its first digits.

We write to denote , the fractional expansion of . If has two different occurrences within at positions and then

Therefore, if and are both no greater than , the pairs and count for . For indices and of which correspond to elements of this is the case:

Then, we are able to give a lower bound for , by taking all possible pairs of occurrences of every word which satisfies the condition :

In the third step, we used the identity

with . Thus,

and the last expression diverges as because is squared in the numerator but linear in the denominator and is insignificant with respect to . This concludes the proof that the sequence of the fractional parts of does not have Poissonian pair correlations.

We now explain how the proof extends to for any given constant in . Take . Then can be written as a concatenation

where each is a -affine necklace. That means that for every , there exists a suitable tuple such that

We take and and we prove that the sequence diverges as . As we did for , it is possible to give a lower bound for by counting occurrences within of words of length .

Fix a non-negative integer . By Lemma 6, there are two possibilities: either maps the subspace of even vectors to itself, or it maps it to the set of vectors beginning with zero. In the first case, we can replicate essentially verbatim the procedure we followed for to get a lower bound for . In the second case, we have to slightly alter the argument: is redefined to be the index of such that matches the first letter of , and is redefined to be the number of ones in . Despite these modifications, we reach the same lower bound for . Since it diverges as we conclude that the sequence of the fractional parts of does not have Poissonian pair correlations.

Acknowledgements

The authors are members of the Laboratoire International Associé SINFIN, Université Paris Diderot-CNRS/Universidad de Buenos Aires-CONICET).

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