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

TL;DR

This paper refines the lower bound for the star discrepancy in three dimensions, showing it grows at least as fast as a logarithmic power with a smaller exponent than previously established.

Contribution

It improves the known lower bound exponent for the three-dimensional star discrepancy, reducing the previously established constant to approximately 0.01736.

Findings

Lower bound exponent for star discrepancy is less than 0.01736.

The result confirms the growth rate of discrepancy with a smaller exponent.

The methods build upon and refine previous discrepancy bounds.

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 $D^{\ast}_N$, is bounded from below by $D_{N}^{\ast}\geq c (\log N)^{1+\eta}$, 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 $\eta<1/(32+4\sqrt{41})\approx 0.017357\ldots$