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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.

L p -discrepancy of the symmetrized van der Corput sequence