Optimal $L_p$-discrepancy bounds for second order digital sequences

TL;DR

This paper proves that order 2 digital $(t,d)$-sequences over the finite field with two elements achieve the optimal order of $L_p$-discrepancy for all $p$ in (1,∞), solving a long-standing problem.

Contribution

It provides a complete solution showing that these digital sequences attain the best possible $L_p$-discrepancy bounds for all finite $p > 1$.

Findings

Order 2 digital $(t,d)$-sequences achieve optimal $L_p$-discrepancy for all $p eq 1$.

The results unify discrepancy bounds across all $p$ in (1,∞).

This advances the understanding of distribution irregularities of digital sequences.

Abstract

The $L_p$-discrepancy is a quantitative measure for the irregularity of distribution modulo one of infinite sequences. In 1986 Proinov proved for all $p>1$ a lower bound for the $L_p$-discrepancy of general infinite sequences in the $d$-dimensional unit cube, but it remained an open question whether this lower bound is best possible in the order of magnitude until recently. In 2014 Dick and Pillichshammer gave a first construction of an infinite sequence whose order of $L_2$-discrepancy matches the lower bound of Proinov. Here we give a complete solution to this problem for all finite $p > 1$. We consider so-called order $2$ digital $(t,d)$-sequences over the finite field with two elements and show that such sequences achieve the optimal order of $L_p$-discrepancy simultaneously for all $p \in (1,\infty)$.