On pair correlation and discrepancy

Abstract: We say that a sequence ${x_n}{n \geq 1}$ in $[0,1)$ has Poissonian pair correlations if \begin{equation*} \lim{N \rightarrow \infty} \frac{1}{N} # \left{ 1 \leq l \neq m \leq N \, : \, \left\lVert x_l-x_m \right\rVert < \frac{s}{N} \right} = 2s \end{equation*} for all $s>0$. 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 ${x_n}_{n \geq 1}$. As an easy consequence it follows that every sequence with Poissonian pair correlations is uniformly distributed in $[0,1)$.