Metrical star discrepancy bounds for lacunary subsequences of digital Kronecker-sequences and polynomial tractability

TL;DR

This paper establishes new discrepancy bounds for lacunary subsequences of digital Kronecker sequences, improving understanding of their distribution properties and potential for quasi-Monte Carlo integration in high dimensions.

Contribution

It provides the first explicit discrepancy bounds for lacunary subsequences of digital Kronecker sequences, extending previous results from classical to digital sequences.

Findings

Discrepancy bounds of C √(d (log d)/N) for digital Kronecker sequences

Extension of lacunary subsequence discrepancy results to digital sequences

Improved understanding of distribution properties of digital sequences in high dimensions

Abstract

The star discrepancy $D_N^*(\mathcal{P})$ is a quantitative measure for the irregularity of distribution of a finite point set $\mathcal{P}$ in the multi-dimensional unit cube which is intimately related to the integration error of quasi-Monte Carlo algorithms. It is known that for every integer $N \ge 2$ there are point sets $\mathcal{P}$ in $[0,1)^d$ with $|\mathcal{P}|=N$ and $D_N^*(\mathcal{P}) =O((\log N)^{d-1}/N)$. However, for small $N$ compared to the dimension $d$ this asymptotically excellent bound is useless (e.g. for $N \le {\rm e}^{d-1}$). In 2001 it has been shown by Heinrich, Novak, Wasilkowski and Wo\'{z}niakowski that for every integer $N \ge 2$ there exist point sets $\mathcal{P}$ in $[0,1)^d$ with $|\mathcal{P}|=N$ and $D_N^*(\mathcal{P}) \le C \sqrt{d/N}$. Although not optimal in an asymptotic sense in $N$, this upper bound has a much better (and even optimal) dependence on the dimension $d$. Unfortunately the result by Heinrich et al. and also later variants thereof by other authors are pure existence results and until now no explicit construction of point sets with the above properties are known. Quite recently L\"obbe studied lacunary subsequences of Kronecker's $(n \boldsymbol{\alpha})$-sequence and showed a metrical discrepancy bound of the form $C \sqrt{d (\log d)/N}$ with implied absolute constant $C>0$ independent of $N$ and $d$. In this paper we show a corresponding result for digital Kronecker sequences, which are a non-archimedean analog of classical Kronecker sequences.