On the upper bound of the $L_p$ discrepancy of Halton's sequence and the Central Limit Theorem for Hammersley's net

TL;DR

This paper establishes the optimal order of the $L_p$ discrepancy for Halton's sequence and proves a Central Limit Theorem for Hammersley's net, advancing understanding of their distributional properties in high-dimensional uniformity.

Contribution

It determines the minimal order of magnitude for the $L_p$ discrepancy of Halton's sequence and proves a CLT for Hammersley's net, using $p$-adic logarithmic forms.

Findings

$L_p$ discrepancy of Halton's sequence is $O(N^{-1} ext{log}^{s/2} N)$

Central Limit Theorem holds for Hammersley's net discrepancy

Identifies the smallest possible order of $L_p$ discrepancy

Abstract

Let $(H_s(n))_{n \geq 1}$ be an $s-$dimensional Halton's sequence, and let ${\mathcal{H}}_{s+1,N}=(H_s(n),n/N)_{n=0}^{N-1}$ be the $s+1-$dimensional Hammersley point set. Let $D(\mathbf{x},(H_n)_{n=0}^{N-1} )$ be the local discrepancy of $(H_n)_{n=0}^{N-1}$, and let $D_{s,p} ( (H_n)_{n=0}^{N-1}) $ be the $L_p$ discrepancy of $(H_n)_{n=0}^{N-1} $. It is known that $\limsup_{N \to \infty} N (\log N)^{-s/2} D_{s,p} (H_s(N))_{n=0}^{N-1} >0$. In this paper, we prove that $$D_{s,p} ((H_s(N))_{n=0}^{N-1}) = O(N^{-1} \log^{s/2} N) \quad {\rm for} \; \; N \to \infty.$$ I.e., we found the smallest possible order of magnitude of $L_p$ discrepancy of Halton's sequence. Then we prove the Central Limit Theorem for Hammersley net : \begin{equation}\nonumber N^{-1} D(\bar{\mathbf{x}},\mathcal{H}_{s+1,N} )/ D_{s+1,2}(\mathcal{H}_{s+1,N}) \stackrel{w}{\rightarrow} \mathcal{N}(0,1), \end{equation} where $\bar{\mathbf{x}}$ is a uniformly distributed random variable in $[0,1]^{s+1}$. The main tool is the theorem on $p$-adic logarithmic forms.