Discrepancy bounds for infinite-dimensional order two digital sequences over $\mathbb{F}_2$

TL;DR

This paper constructs explicit infinite-dimensional digital sequences over _2 with provably optimal discrepancy bounds in the _q norm, improving understanding of their uniform distribution properties.

Contribution

It provides explicit constructions of digital sequences with discrepancy bounds in _q norms, extending the theory of low-discrepancy sequences in infinite dimensions.

Findings

Discrepancy bounds of order N^{-1} with polynomial factors in _q norms.

For N=2^m, discrepancy is of order m^{(s-1)/2} 2^{-m}.

Sequences achieve near-optimal uniform distribution in infinite dimensions.

Abstract

In this paper we provide explicit constructions of digital sequences over the finite field of order 2 in the infinite dimensional unit cube whose first $N$ points projected onto the first $s$ coordinates have $\mathcal{L}_q$ discrepancy bounded by $r^{3/2-1/q} \sqrt{m_1^{s-1} + m_2^{s-1} + \cdots + m_r^{s-1}} N^{-1}$ for all $N = 2^{m_1} + 2^{m_2} + \cdots + 2^{m_r} \ge 2$ and $2 \le q < \infty$. In particular we have for $N = 2^m$ that the $\mathcal{L}_q$ discrepancy is of order $m^{(s-1)/2} 2^{-m}$ for all $2 \le q < \infty$.