On pair correlation and discrepancy

By Sigrid Grepstad, G. Larcher. 2016

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

Subfields: Algebra and Number Theory, Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/4fc0f8198660958d71f23dd12c6967c335baa85c