On the distribution of the van der Corput sequence in arbitrary base

TL;DR

This paper establishes a central limit theorem and large deviation bounds for the van der Corput sequence in any base, analyzing its distribution and discrepancy using Fourier analysis and probabilistic methods.

Contribution

It extends the probabilistic analysis of the van der Corput sequence to arbitrary bases, providing explicit error bounds and large deviation results.

Findings

Central limit theorem for sums over the van der Corput sequence

Explicit error bounds for distribution approximations

Results on the L^p discrepancy of the sequence

Abstract

A central limit theorem with explicit error bound, and a large deviation result are proved for a sequence of weakly dependent random variables of a special form. As a corollary, under certain conditions on the function $f: [0,1] \to \mathbb{R}$ a central limit theorem and a large deviation result are obtained for the sum $\sum_{n=0}^{N-1} f(x_n)$, where $x_n$ is the base $b$ van der Corput sequence for an arbitrary integer $b \ge 2$. Similar results are also proved for the $L^p$ discrepancy of the same sequence for $1 \le p < \infty$. The main methods used in the proofs are the Berry-Esseen theorem and Fourier analysis.