Low Discrepancy Digital Kronecker-Van der Corput Sequences

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.