Optimal Polynomial Tractability Exponents for the Inverse Star Discrepancy

The inverse of the star discrepancy $n^\ast(d, \varepsilon)$ satisfies $$d \varepsilon^{-1} \lesssim n^{\ast}(d,\varepsilon)\lesssim d\varepsilon^{-2}$$ for all $d \in \mathbb{N}$ and $0 < \varepsilon < \varepsilon_0$. The upper bound was shown by Heinrich, Novak, Wasilkowski and Wo\'{z}niakowski (2001) and the lower bound was shown by Hinrichs (2004). A proof of this lower bound using elementary arguments was shown by Steinerberger (2023). These bounds show that the inverse of the star discrepancy depends linearly on the dimension, but the exact exponent of $\varepsilon^{-1}$ has remained open. In this paper we prove a lower bound which shows that the exponent $2$ of $\varepsilon^{-1}$ in the upper bound cannot be improved. More precisely, we show that for every $0<\alpha<1$ and fixed $0<A\le B$, there are constants $c_{\alpha,B}>0$ and $\varepsilon_{\alpha,A}>0$ such that for every integer $d$ satisfying $$A\varepsilon^{-\alpha}\le d\le B\varepsilon^{-\alpha}$$ we have $$n^{\ast}(d,\varepsilon) \ge c_{\alpha,B}\,d\,\varepsilon^{-(2-\alpha)},$$ for all $0 < \varepsilon < \varepsilon_{\alpha, A}$. Along these polynomial strips the right-hand side is of order $\varepsilon^{-2}$. It follows that every uniform polynomial upper estimate $n^{\ast}(d,\varepsilon)\le C d^q\varepsilon^{-p}$ must satisfy $p \ge 2$. Together with the lower bound of Hinrichs (2004), which forces $q \ge 1$, this shows that the exponents $p=2$ and $q=1$ in the Heinrich--Novak--Wasilkowski--Wo\'{z}niakowski upper bound are individually optimal.