Discrepancy estimates for sequences: new results and open problems

In this paper we give an overview of recent results on (upper and lower) discrepancy estimates for (concrete) sequences in the unit-cube. In particular we also give an overview of discrepancy estimates for certain classes of hybrid sequences. Here by a hybrid sequence we understand an $(s+t)$-dimensional sequence which is a combination of an $s$-dimensional sequence of a certain type (e.g. Kronecker-, Niederreiter-, Halton-,... type) and a $t$-dimensional sequence of another type. The analysis of the discrepancy of hybrid sequences (and of their components) is a rather current and vivid branch of research. We give a collection of some challenging open problems on this topic.Comment: 17 page

On pair correlation and discrepancy

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 $[0,1)$.Comment: To appear in Archiv der Mathemati