Low Discrepancy Digital Kronecker-Van der Corput Sequences

TL;DR

This paper establishes a connection between Diophantine approximation over function fields and low discrepancy hybrid sequences constructed from digital Kronecker and Van der Corput sequences, providing explicit conditions for low discrepancy.

Contribution

It introduces a method to construct low discrepancy hybrid sequences using Laurent series and irreducible polynomials, linking discrepancy properties to the $t$-adic Littlewood Conjecture.

Findings

Hybrid sequences meet low discrepancy criteria under certain conditions.

Counterexamples to the $t$-adic Littlewood Conjecture lead to low discrepancy sequences.

Explicit construction of sequences with proven discrepancy properties.

Abstract

The discrepancy of a sequence measures how quickly it approaches a uniform distribution. Given a natural number $d$, any collection of one-dimensional so-called low discrepancy sequences $\left\{S_i:1\le i \le d\right\}$ can be concatenated to create a $d$-dimensional $\textit{hybrid sequence}$ $(S_1,\dots,S_d)$. Since their introduction by Spanier in 1995, many connections between the discrepancy of a hybrid sequence and the discrepancy of its component sequences have been discovered. However, a proof that a hybrid sequence is capable of being low discrepancy has remained elusive. This paper remedies this by providing an explicit connection between Diophantine approximation over function fields and two dimensional low discrepancy hybrid sequences. Specifically, let $\mathbb{F}_q$ be the finite field of cardinality $q$. It is shown that some real numbered hybrid sequence $\mathbf{H}(\Theta(t),P(t)):=\textbf{H}(\Theta,P)$ built from the digital Kronecker sequence associated to a Laurent series $\Theta(t)\in\mathbb{F}_q((t^{-1}))$ and the digital Van der Corput sequence associated to an irreducible polynomial $P(t)\in\mathbb{F}_q[t]$ meets the above property. More precisely, if $\Theta(t)$ is a counterexample to the so called $t$$\textit{-adic Littlewood Conjecture}$ ($t$-$LC$), then another Laurent series $\Phi(t)\in\mathbb{F}_q((t^{-1}))$ induced from $\Theta(t)$ and $P(t)$ can be constructed so that $\mathbf{H}(\Phi,P)$ is low discrepancy. Such counterexamples to $t$-$LC$ are known over a number of finite fields by, on the one hand, Adiceam, Nesharim and Lunnon, and on the other, by Garrett and the author.