On the Gap between Hereditary Discrepancy and the Determinant Lower Bound
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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, √(log m)}. On the other hand, building on work of Matoušek [Proceedings of the AMS, 2013], recently Jiang and Reis [SOSA, 2022] 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 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.
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