Optimal periodic L_2-discrepancy and diaphony bounds for higher order digital sequences

12/12/2022

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by   Friedrich Pillichshammer, et al.

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We present an explicit construction of infinite sequences of points (x_0,x_1, x_2, …) in the d-dimensional unit-cube whose periodic L_2-discrepancy satisfies L_2,N^ per({x_0,x_1,…, x_N-1}) ≤ C_d N^-1 (log N)^d/2 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 L_2-discrepancy is equivalent to P. Zinterhof's diaphony the result also applies to this alternative quantitative measure for the irregularity of distribution.