AI 中文总结
研究逆星偏差的最优多项式易处理性指数，通过证明下界表明上界中\(\varepsilon^{-1}\)的指数2不能改进，结合已有下界得出海因里希 - 诺瓦克 - 瓦西科夫斯基上界中指数\(p = 2\)和\(q = 1\)分别最优。
AI 中文摘要
星偏差 \(n^\ast(d, \varepsilon)\) 的逆满足对于所有 \(d \in \mathbb{N}\) 和 \(0 < \varepsilon < \varepsilon_0\)，有 \(d \varepsilon^{-1} \lesssim n^{\ast}(d,\varepsilon)\lesssim d\varepsilon^{-2}\)。上界由海因里希、诺瓦克、瓦西科夫斯基和沃兹尼亚科夫斯基（2001 年）给出，下界由欣里希斯（2004 年）给出，施泰纳伯格（2023 年）用基本论证给出了该下界的证明。本文证明了一个下界，表明上界中 \(\varepsilon^{-1}\) 的指数 2 不能改进。具体而言，对于每个 \(0<\alpha<1\) 和固定的 \(0<A\le B\)，存在常数 \(c_{\alpha,B}>0\) 和 \(\varepsilon_{\alpha,A}>0\)，使得对于满足 \(A\varepsilon^{-\alpha}\le d\le B\varepsilon^{-\alpha}\) 的每个整数 \(d\)，有 \(n^{\ast}(d,\varepsilon) \ge c_{\alpha,B}\,d\,\varepsilon^{-(2 - \alpha)}\)，对于所有 \(0 < \varepsilon < \varepsilon_{\alpha, A}\)。由此得出每个一致多项式上界估计 \(n^{\ast}(d,\varepsilon)\le C d^q\varepsilon^{-p}\) 必须满足 \(p \ge 2\)。结合欣里希斯（2004 年）的下界，这表明海因里希 - 诺瓦克 - 瓦西科夫斯基 - 沃兹尼亚科夫斯基上界中的指数 \(p = 2\) 和 \(q = 1\) 分别是最优的。
英文摘要
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 Woźniakowski (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 $0<α<1$ and fixed $0<A\le B$, there exist constants $c_{α,B}>0$ and $\varepsilon_{α,A}>0$ such that, for every $0<\varepsilon<\varepsilon_{α,A}$ and every integer $d$ satisfying \[ A\varepsilon^{-α}\le d\le B\varepsilon^{-α}, \] one has \[ n^\ast(d,\varepsilon) \ge c_{α,B}\,d\,\varepsilon^{-(2-α)}. \] 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--Woźniakowski upper bound are individually optimal. In particular, the optimal exponent $p^\ast = 2$, thereby proving the Novak--Woźniakowski conjecture.
Comments11 pages, 1 figure
