On the upper bound of the $L_2$-discrepancy of Halton's sequence

TL;DR

This paper establishes that the $L_2$-discrepancy of 2-dimensional Halton sequences grows at most on the order of log N, identifying the minimal possible growth rate for such sequences.

Contribution

It proves an upper bound of order log N for the $L_2$-discrepancy of 2D Halton sequences, matching the known lower bound, thus determining its exact order of magnitude.

Findings

$L_2$-discrepancy of Halton sequences is $O( ext{log } N)$

Established the minimal possible order of growth for discrepancy

Used $p$-adic logarithm linear forms in the proof

Abstract

Let $(H(n))_{n \geq 0} $ be a $2-$dimensional Halton's sequence. Let $D_{2} ( (H(n))_{n=0}^{N-1}) $ be the $L_2$-discrepancy of $ (H_n)_{n=0}^{N-1} $. It is known that $\limsup_{N \to \infty } (\log N)^{-1} D_{2} ( H(n) )_{n=0}^{N-1} >0$. In this paper, we prove that $$D_{2} (( H(n) )_{n=0}^{N-1}) =O( \log N) \quad {\rm for} \; \; N \to \infty ,$$ i.e., we found the smallest possible order of magnitude of $L_2$-discrepancy of a 2-dimensional Halton's sequence. The main tool is the theorem on linear forms in the $p$-adic logarithm.