Optimal periodic L₂-discrepancy and diaphony bounds for higher order digital sequences

2022/12/12 by Friedrich Pillichshammer, Pillichshammer, Friedrich
Mathematics · #11K06 #11K38 #11K45 #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT) #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2212.05747

openalex publication_date 2022/12/12 · openalex created_date 2022/12/26 · openalex updated_date 2026/08/01

Abstract

We present an explicit construction of infinite sequences of points (\boldsymbolx₀,\boldsymbolx₁, \boldsymbolx₂, …) in the d-dimensional unit-cube whose periodic L₂-discrepancy satisfies L_2,N^{\rm per}(\\boldsymbolx₀,\boldsymbolx₁,…, \boldsymbolx_N-1\) ≤ C_d N^-1 (log N)^d/2 for all N ≥ 2, where the factor C_d > 0 depends only on the dimension d. The construction is based on higher order digital sequences as introduced by J. Dick in the year 2008. The result is best possible in the order of magnitude in N according to a Roth-type lower bound shown first by P.D. Proinov. Since the periodic L₂-discrepancy is equivalent to P. Zinterhof's diaphony the result also applies to this alternative quantitative measure for the irregularity of distribution.

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