论文标题
在$ l_2 $ -discrepancy的上限上
On the upper bound of the $L_2$-discrepancy of Halton's sequence
论文作者
论文摘要
令$（h（n））_ {n \ geq 0} $为$ 2- $ dimensional halton的序列。令$ d_ {2}（（h（n））_ {n = 0}^{n-1}）$为$ l_2 $ -discRepancy $（h_n）_ {n = 0}^n = 0}^{n-1} $。众所周知，$ \ limsup_ {n \ to \ infty}（\ log n）^{ - 1} d_ {2}（h（n））_ {n = 0}^{n-1}> 0 $。在本文中，我们证明$$ d_ {2}（（h（n））_ {n = 0}^{n-1}）= o（\ log n）\ quad {\ rm for} \; \; n \ to \ infty，$$，即，我们发现二维Halton的序列的最小数量级为$ l_2 $ -disccrepancy。主要工具是$ p $ -Adic对数中线性形式的定理。
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.
