Some negative results related to Poissonian pair correlation problems

Abstract

We say that a sequence (xn) n∈N in [0, 1) has Poissonian pair correlations iffor every s ≥ 0. The aim of this article is twofold. First, we will establish a gap theorem which allows to deduce that a sequence (xn) n∈N of real numbers in [0, 1) having a certain weak gap structure, cannot have Poissonian pair correlations. This result covers a broad class of sequences, e.g., Kronecker sequences, the van der Corput sequence and in more general LS-sequences of points and digital (t, 1)-sequences. Additionally, this theorem enables us to derive negative pair correlation properties for sequences of the form ({anα}) n∈N , where (an) n∈N is a strictly increasing sequence of integers with maximal order of additive energy, a notion that plays an important role in many fields, e.g., additive combinatorics, and is strongly connected to Poissonian pair correlation problems. These statements are not only metrical results, but hold for all possible choices of α. Second, in this note we study the pair correlation statistics for sequences of the form, xn = {b n α}, n = 1, 2, 3, . . ., with an integer b ≥ 2, where we choose α as the Stoneham number and as an infinite de Bruijn word. We will prove that both instances fail to have the Poissonian property. Throughout this article {·} denotes the fractional part of a real number. arXiv:1803.05236v2 [math.NT]

Pair correlation of sequences with maximal additive energy

“…The result was already proven by the first author for d = 1 in a previous work, see [9]. This case can also be recovered by Theorem 1 in [10].…”

7. On pair correlation of sequences

“…In fact, a deeper investigation reveals that the Three-Gap-Theorem is indeed a valid argument to deduce the result on the PPC strucuture of the Kronecker sequence. It is an immediate consequence of the following theorem, which was proven in [18], in combination with the Three-Gap-Theorem. Theorem 1.…”

“…However, it was shown by Pirsic and Stockinger in [22], that for α the Champernowne number, the sequence ({2 n α}) n≥1 does not have PPC. Also for further concrete examples like Stoneham-numbers or infinite de Bruijn-words, the sequence ({2 n α}) n≥1 does not have PPC (see [18]).…”

The Champernowne constant is not Poissonian

“…Note that the other direction is not necessarily correct. For instance the Kronecker sequence ({nα}) n∈N , does not have this property for any real α; a fact that can be argued by a continued fractions argument or by the main theorem in [12] in combination with the famous Three Gap Theorem, see [17]. Poissonian pair correlation is a typical property of a sequence.…”