On an explicit lower bound for the star discrepancy in three dimensions

Abstract

Following a result of D.~Bylik and M.T.~Lacey from 2008 it is known that there exists an absolute constant $\eta >0$ such that the (unnormalized) $L^{\infty }$-norm of the three-dimensional discrepancy function, i.e, the (unnormalized) star discrepancy $D_N^{\ast }$, is bounded from below by $D_N^{\ast }\ge c(\log N{)}^{1+\eta }$, for all $N\in \mathbb{N}$ sufficiently large, where $c>0$ is some constant independent of $N$. This paper builds upon their methods to verify that the above result holds with $\eta <1/(32+4\sqrt{41})\approx 0.017357\dots$

arXiv comment15 pages, 1 figure; includes minor corrections as well as an additional brief section on the littlewood-paley inequalities

Cited by in corpus (1)

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