On the Gap Between Hereditary Discrepancy and the Determinant Lower Bound

.The determinant lower bound of Lovász, Spencer, and Vesztergombi [European J. Combin., 7 (1986), pp. 151–160] is a general way to prove lower bounds on the hereditary discrepancy of a set system. In their paper, Lovász, Spencer, and Vesztergombi asked if hereditary discrepancy can also be bounded from above by a function of the determinant lower bound. This was answered in the negative by Hoffman, and the largest known multiplicative gap between the two quantities for a set system of \(m\) subsets of a universe of size \(n\) is on the order of \(max \{log n,√log m\}\). On the other hand, building upon work of Matoušek [Proc. Amer. Math. Soc., 141 (2013), pp. 451–460], Jiang and Reis [in Proceedings of the Symposium on Simplicity in Algorithms (SOSA), SIAM, Philadelphia, 2022, pp. 308–313] showed that this gap is always bounded up to constants by \(√log (m)log (n)\). This is tight when \(m\) is polynomial in \(n\) but leaves open the case of large \(m\). We show that the bound of Jiang and Reis is tight for nearly the entire range of \(m\). Our proof amplifies the discrepancy lower bounds of a set system derived from the discrete Haar basis via Kronecker products.