On pair correlation and discrepancy

We say that a sequence xnn≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( x_n\right) _{n \ge 1}$$\end{document} in [0, 1) has Poissonian pair correlations if limN→∞1N#1≤l≠m≤N:|xl-xm|<sN=2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{N \rightarrow \infty } \frac{1}{N} \# \left\{ 1 \le l \ne m \le N{:}\,\left||x_l-x_m\right|| < \frac{s}{N} \right\} = 2s \end{aligned}$$\end{document}for all s>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>0$$\end{document}. In this note we show that if the convergence in the above expression is—in a certain sense—fast, then this implies a small discrepancy for the sequence xnn≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( x_n\right) _{n \ge 1}$$\end{document}. As an easy consequence it follows that every sequence with Poissonian pair correlations is uniformly distributed in [0, 1).


Introduction.
The concept of Poissonian pair correlations for a sequence (x n ) n≥1 in [0, 1) was introduced by Rudnick and Sarnak [5], and has been intensively studied by several authors over the last years (see, for instance, [2,3,[6][7][8]). Let · denote distance to the nearest integer. We say that a sequence (x n ) n≥1 of real numbers in [0, 1) has Poissonian pair correlations if for every s > 0.
In this note we are concerned with the relation between the Poissonian pair correlation property and the notion of uniform distribution. We say that the sequence (x n ) n≥1 is uniformly distributed, or equidistributed, It is well known that uniform distribution does not necessarily imply Poissonian pair correlations. One example confirming this is the Kronecker sequence ({nα}) n≥1 , which is uniformly distributed for every irrational α, but does not have Poissonian pair correlations for any value of α. Whether the converse implication holds has until recently remained an open question: is every sequence in [0, 1) with Poissonian pair correlations uniformly distributed? We answer this question in the affirmative by establishing a quantitative result connecting the speed of convergence in (1.1) to the stardiscrepancy D * N of the sequence. We recall that the star-discrepancy D * N of (x n ) n≥1 is defined as where A N ([0, a)) := #{1 ≤ n ≤ N : x n ∈ [0, a)}, and that (x n ) n≥1 is uniformly distributed in [0, 1) if and only if lim N →∞ D * N = 0 (see, for example, [4]). The main result of this paper is the following. 1), and suppose that there exists a function F : N × N → R + which is monotonically increasing in its first argument, and which satisfies max s=1,...,K 1 2s for all N ∈ N and all K ≤ N/2. One can then find an integer N 0 > 0 such that for N ∈ N, N ≥ N 0 , and arbitrary K satisfying N is the star-discrepancy of (x n ) n≥1 . The next result is an easy consequence of Theorem 1.1.

Proof
Aiming for a proof by contradiction, we assume that ND * N > H(N, K) for infinitely many pairs (N, K). That is, there exist integers 1 < N 1 < N 2 < · · · and corresponding integers K 1 , K 2 , . . . satisfying (1.3), as well as real numbers for every j. We assume in what follows that (2.1) holds (the case when (2.2) holds is treated analogously). Note that (2.1) implies We now consider the distribution of the points x n into subintervals of [0, 1) of length K/N . Let for i = 0, 1, . . . , N/K − 1, and let

Moreover, for arbitrary positive integers l, let
A l := A l mod( N/K +1) . We have that

If we introduce the notation
Thus, we get .
and thus We have To see this, assume to the contrary that Z 1 and Z L − (L − 1)Z L−1 /L are all less than 2Z K /(K + 1). Then by successive insertions we get the contradiction Z K < Z K . Hence, we have Let us now estimate min x1,...,xN where the minimum on the right-hand side is taken over all possible values of A 0 , A 1 , . . . , A N/K provided that the points x 1 , . . . , x N satisfy (2.1). By Vol. 109 (2017) On pair correlation and discrepancy 147 definition, we have A 0 + · · · + A N/K = N . Introducing the notation G i = A i + A i+1 + · · · + A i+K−1 , we thus get Moreover, by invoking condition (2.1) on the distribution of x 1 , . . . , x N , we have (2.9) and consequently We get min x1,...,xN where the minimum on the right-hand side is taken over all positive reals G 0 , G 1 , . . . , G N/K satisfying (2.8)-(2.10). It is an easy exercise to verify that this minimum is attained when Note that since K ≤ N 2/5 and H ≥ 5N 4/5 , we have K 2 ≤ H/5, and hence by (2.3) both the numerator and the denominator of these G i are positive. Thus, we get which is a contradiction. Thus, our assumption (2.1) must be incorrect, and the proof of Theorem 1.1 is complete. (Note that the last inequality above is trivially true if N 2 /K ≤ NF (K 2 , N); in the opposite case we have K < N/F (K 2 , N), and by the condition (1.3) imposed on K, we then get K ≥ N 2/5 /2, and consequently N 2 /K ≤ 2N 8/5 .)