2018/06/29 by Mordechay B. Levin, Levin, Mordechay B.
Mathematics · #11K38 #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1806.11498
openalex publication_date 2018/06/29 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
Let (Hs(n))n ≥ 1 be an s-dimensional Halton's sequence, and let Hs+1,N=(Hs(n),n/N)n=0N-1 be the s+1-dimensional Hammersley point set. Let D(x,(Hn)n=0N-1 ) be the local discrepancy of (Hn)n=0N-1, and let Ds,p ( (Hn)n=0N-1) be the Lp discrepancy of (Hn)n=0N-1 . It is known that \limsupN → ∞ N (log N)-s/2 Ds,p (Hs(N))n=0N-1 >0. In this paper, we prove that Ds,p ((Hs(N))n=0N-1) = O(N-1 logs/2 N) \rm for N → ∞. I.e., we found the smallest possible order of magnitude of Lp discrepancy of Halton's sequence. Then we prove the Central Limit Theorem for Hammersley net : N-1 D(x,Hs+1,N )/ Ds+1,2(Hs+1,N) \stackrelw→ N(0,1), where x is a uniformly distributed random variable in [0,1]s+1. The main tool is the theorem on p-adic logarithmic forms.