Optimal periodic $L_2$-discrepancy and diaphony bounds for higher order digital sequences

TL;DR

This paper constructs explicit higher order digital sequences in high dimensions that achieve optimal bounds on periodic $L_2$-discrepancy and diaphony, improving understanding of distribution irregularities.

Contribution

It provides the first explicit construction of sequences with optimal order bounds on periodic $L_2$-discrepancy using higher order digital sequences.

Findings

Achieves discrepancy bound of $C_d N^{-1} ( ext{log} N)^{d/2}$ for all $N \\ge 2$

Construction based on higher order digital sequences by J. Dick (2008)

Bound is optimal in order of magnitude according to Roth-type lower bounds

Abstract

We present an explicit construction of infinite sequences of points $(\boldsymbol{x}_0,\boldsymbol{x}_1, \boldsymbol{x}_2, \ldots)$ in the $d$-dimensional unit-cube whose periodic $L_2$-discrepancy satisfies $$L_{2,N}^{{\rm per}}(\{\boldsymbol{x}_0,\boldsymbol{x}_1,\ldots, \boldsymbol{x}_{N-1}\}) \le C_d N^{-1} (\log N)^{d/2} \quad \mbox{for all } N \ge 2,$$ where the factor $C_d > 0$ depends only on the dimension $d$. The construction is based on higher order digital sequences as introduced by J. Dick in the year 2008. The result is best possible in the order of magnitude in $N$ according to a Roth-type lower bound shown first by P.D. Proinov. Since the periodic $L_2$-discrepancy is equivalent to P. Zinterhof's diaphony the result also applies to this alternative quantitative measure for the irregularity of distribution.