A Proof of the Novak--Wo\'{z}niakowski Conjecture: Optimal Polynomial Tractability Exponents for the Inverse Star Discrepancy

Abstract

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 established by Heinrich, Novak, Wasilkowski and Wo\'{z}niakowski (2001), while the lower bound is due to Hinrichs (2004). Steinerberger (2023) subsequently gave an elementary proof of the latter result. These bounds imply that the inverse of the star discrepancy depends linearly on the dimension, but the exact exponent of $\varepsilon^{-1}$ had 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, for every $0<\alpha<1$ and fixed $0 0$ and $\varepsilon_{\alpha,A}>0$ such that, for every $0<\varepsilon<\varepsilon_{\alpha,A}$ and every integer $d$ satisfying \[ A\varepsilon^{-\alpha}\le d\le B\varepsilon^{-\alpha}, \] one has \[ n^\ast(d,\varepsilon) \ge c_{\alpha,B}\,d\,\varepsilon^{-(2-\alpha)}. \] Along these polynomial strips the right-hand side is of order $\varepsilon^{-2}$. Consequently, every uniform polynomial upper estimate $n^{\ast}(d,\varepsilon)\le C d^q\varepsilon^{-p}$ must satisfy $p\ge2$. Together with the lower bound of Hinrichs (2004), which forces $q\ge1$, this proves that the exponents $p=2$ and $q=1$ in the Heinrich--Novak--Wasilkowski--Wo\'{z}niakowski upper bound are individually optimal. In particular, the optimal exponent $p^\ast = 2$, thereby proving the Novak--Wo\'{z}niakowski conjecture.