Niederreiter and Halton sequences are two prominent classes of higher-dimensional sequences which are widely used in practice for numerical integration methods because of their excellent distribution qualities. In this paper we show that these sequences—even though they are uniformly distributed—fail to satisfy the stronger property of Poissonian pair correlations. This extends already established results for one-dimensional sequences and confirms a conjecture of Larcher and Stockinger who hypothesized that the Halton sequences are not Poissonian. The proofs rely on a general tool which identifies a specific regularity of a sequence to be sufficient for not having Poissonian pair correlations.

Niederreiter和Halton序列是两大类的高维序列，由于其出色的分布质量而在实践中广泛用于数值积分方法。在本文中，我们证明了这些序列（即使它们是均匀分布的）也无法满足泊松对相关的更强特性。这扩展了已经为一维序列建立的结果，并证实了Larcher和Stockinger的猜想，他们推测Halton序列不是泊松序列。证明依赖于一个通用工具，该工具识别序列的特定规律性足以使其不具有泊松对相关性。