The determinant lower bound of Lovasz, Spencer, and Vesztergombi [European Journal of Combinatorics, 1986] is a powerful general way to prove lower bounds on the hereditary discrepancy of a set system. In their paper, Lovasz, Spencer, and Vesztergombi asked if hereditary discrepancy can also be bounded from above by a function of the hereditary discrepancy. This was answered in the negative by Hoffman, and the largest known multiplicative gap between the two quantities for a set system of $m$ substes of a universe of size $n$ is on the order of $\max\{\log n, \sqrt{\log m}\}$. On the other hand, building on work of Matou\v{s}ek [Proceedings of the AMS, 2013], recently Jiang and Reis [SOSA, 2022] showed that this gap is always bounded up to constants by $\sqrt{\log(m)\log(n)}$. This is tight when $m$ is polynomial in $n$, but leaves open what happens for large $m$. We show that the bound of Jiang and Reis is tight for nearly the entire range of $m$. Our proof relies on a technique of amplifying discrepancy via taking Kronecker products, and on discrepancy lower bounds for a set system derived from the discrete Haar basis.

翻译：Lovasz、Spencer和Vesztergombi [European Journal of Combinatorics, 1986] 提出的行列式下界是证明集合系统遗传偏差下界的一种通用的强大方法。在他们的论文中，Lovasz、Spencer和Vesztergombi 曾提出疑问：遗传偏差是否也能被某个函数从上方界定？Hoffman 否定了这一可能性，而对于一个包含大小为 $n$ 的全集中 $m$ 个子集的集合系统，这两个量之间已知的最大乘法间隙量级为 $\max\{\log n, \sqrt{\log m}\}$。另一方面，基于 Matoušek [Proceedings of the AMS, 2013] 的工作，Jiang 和 Reis [SOSA, 2022] 最近表明，这个间隙始终被 $\sqrt{\log(m)\log(n)}$ 常数倍界定。当 $m$ 是 $n$ 的多项式时，这一结果是紧的，但未能解决 $m$ 很大时的情况。我们证明 Jiang 和 Reis 的界在 $m$ 的几乎整个范围内都是紧的。我们的证明依赖于一种通过克罗内克积放大偏差的技术，以及基于离散 Haar 基导出的集合系统的偏差下界。