Abstract
It is well known that the L p -discrepancy for \({p \in [1,\infty]}\) of the van der Corput sequence is of exact order of magnitude \({O((\log N)/N)}\). This however is for \({p \in (1,\infty)}\) not best possible with respect to the lower bounds according to Roth and Proinov. For the case \({p=2}\), it is well known that the symmetrization trick due to Davenport leads to the optimal L 2-discrepancy rate \({O(\sqrt{\log N}/N)}\) for the symmetrized van der Corput sequence. In this note we show that this result holds for all \({p \in [1,\infty)}\). The proof is based on an estimate of the Haar coefficients of the corresponding local discrepancy and on the use of the Littlewood-Paley inequality.
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The authors are supported by the Austrian Science Fund (FWF): Project F5509-N26, which is a part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.
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Kritzinger, R., Pillichshammer, F. L p -discrepancy of the symmetrized van der Corput sequence. Arch. Math. 104, 407–418 (2015). https://doi.org/10.1007/s00013-015-0760-7
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DOI: https://doi.org/10.1007/s00013-015-0760-7