On the lower bound of the discrepancy of Halton's sequence II

2015/07/30 by Mordechay B. Levin, Levin, Mordechay B. · 1 citation
Mathematics · #11K38 #Analytic Number Theory Research #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1507.08529

openalex publication_date 2015/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (H_s(n))_{n ≥ 1} be an s-dimensional generalized Halton's sequence. Let D^*_N be the discrepancy of the sequence (H_s(n) )_{n = 1}^N . It is known that D^*_N =O(ln^s N) as N → ∞ . In this paper, we prove that this estimate is exact. Namely, there exists a constant C(H_s)>0, such that max_{1 ≤ M ≤ N} M D^*_M ≥ C(H_s) log₂^s N \rm for N=2,3,... .

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