Digital nets in dimension two with the optimal order of L_p discrepancy [article]

Ralph Kritzinger, Friedrich Pillichshammer
2018 arXiv   pre-print
We study the L_p discrepancy of two-dimensional digital nets for finite p. In the year 2001 Larcher and Pillichshammer identified a class of digital nets for which the symmetrized version in the sense of Davenport has L_2 discrepancy of the order √( N)/N, which is best possible due to the celebrated result of Roth. However, it remained open whether this discrepancy bound also holds for the original digital nets without any modification. In the present paper we identify nets from the above
more » ... ned class for which the symmetrization is not necessary in order to achieve the optimal order of L_p discrepancy for all p ∈ [1,∞). Our findings are in the spirit of a paper by Bilyk from 2013, who considered the L_2 discrepancy of lattices consisting of the elements (k/N,{k α}) for k=0,1,...,N-1, and who gave Diophantine properties of α which guarantee the optimal order of L_2 discrepancy.
arXiv:1804.04891v1 fatcat:qljasfnoxrg4veoi3sxyshswz4