Pair correlation and equidistribution on manifolds

Abstract

This study is motivated by a series of recent papers that show that, if a given deterministic sequence in the unit interval has a Poisson pair correlation function, then the sequence is uniformly distributed. Analogous results have been proved for point sequences on higher-dimensional tori. The purpose of this paper is to describe a simple statistical argument that explains this observation and furthermore permits a generalisation to bounded Euclidean domains as well as compact Riemannian manifolds.

Weak Poissonian Correlations

“…To mention an example, it was an open problem within the theoretical setup to determine the relation of Poissonian pair correlations with uniform distribution. It has been recently shown that any sequence (x n ) n∈N with PPC is also uniformly distributed mod 1, that is, we have This result was established by Aistleiter, Larcher and Lewko [2] and independently by Grepstad and Larcher [8], while a subsequent proof was also given by Steinerberger [24] and, in a much more general setup, by Marklof [16].…”

“…(where the zero -measure set depends on s). Repeating the argument for all s lying in a dense, countable subset of R + and employing the monotonicity of I N (s) as a function of s, we see that (16) actually holds for all s > 0. Next, if N 1 is an arbitrary integer, we let K 1 be such that B K N < B K+1 and observe that for any s > 0,…”

7. On pair correlation of sequences

“…Of course it makes sense to generalize the concept of PPC to the multidimensional setting. One way to generalize the one-dimensional concept to a multi-dimensioanl setting was defined and discussed in [15] (for a more general analysis of a multi-dimensional PPC concept, we refer to the recent work [21]). Here, we present the definition of [15].…”

Poissonian correlations of higher orders

“…Theorem A was proved independently by Aistleitner, Lachmann and Pausinger [1] and by Grepstad and Larcher [9]. Additional proofs were given later by Steinerberger in [25] and, in a much more general setup, by Marklof [15]. These four proofs are all essentially different.…”