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：Lovász、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差异下界的一种通用强效方法。在他们的论文中，他们提出了一个问题：遗传差异是否也能由某个遗传差异的函数给出上界？Hoffman对此给出了否定答案，并且对于一个包含$m$个子集且全集大小为$n$的集合系统，这两个量之间已知的最大乘法差距的量级为$\max\{\log n, \sqrt{\log m}\}$。另一方面，基于Matoušek [Proceedings of the AMS, 2013] 的工作，Jiang和Reis [SOSA, 2022] 最近证明了这个差距在常数范围内始终以$\sqrt{\log(m)\log(n)}$为界。当$m$是$n$的多项式时，这个界是紧的，但大型$m$的情况仍然未解。我们证明了Jiang和Reis的界在几乎整个$m$的范围内都是紧的。我们的证明依赖于通过克罗内克积放大差异的技术，以及从离散Haar基导出的一个集合系统的差异下界。