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

Abstract

Let $(H(n){)}_{n\ge 0}$ 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 ${\mathrm{limsup}}_{N\to \infty }(\log N{)}^{-1}{D}_2(H(n){)}_{n=0}^{N-1}>0$. In this paper, we prove that $$D_2((H(n){)}_{n=0}^{N-1})=O(\log N)\mathrm{for}N\to \infty ,$$ 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.

arXiv admin note: text overlap with arXiv:1806.11498