  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 $[0,1)$.
