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

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 $\eta<1/(32+4\sqrt{41})\approx 0.017357\ldots$