Lp discrepancy of symmetrized Hammersley point sets

Abstract: 

It is well known that the symmetrization of finite point sets and infinite sequences can lead to the optimal order of Lp discrepancy in the sense of results by Roth, Schmidt and Halász. This method was introduced by Davenport in 1956. It this talk we consider symmetrized Hammersley point sets in dimension two and arbitrary base. To obtain results on their Lp discrepancy, we employ two different approaches:

• Harmonic analysis: We demonstrate how Littlewood Paley theory can be used to prove the optimal order of Lp discrepancy for all p ∈ [1, ∞).
• Number theory: For the L2 discrepancy, it is even possible to obtain an exact formula, which allows to determine the leading constants of the L2 discrepancy. We sketch the elementary proof and compare our results to the currently best known lower and upper bounds on the L2 discrepancy of two-dimensional point sets