On Combinatorial Rectangles with Minimum $\infty$-Discrepancy

TL;DR

This paper studies combinatorial rectangles (matrices with entries ±1) with minimal infinity-discrepancy, providing explicit formulas for small cases and asymptotic estimates for larger matrices, along with a new theoretical framework based on majorization.

Contribution

It introduces a new theory of decreasing criteria on row-sum vectors using majorization, aiding the analysis of minimal discrepancy matrices.

Findings

Explicit formulas for matrices with up to 4 rows and minimal discrepancy

Asymptotic order of magnitude for large matrices with fixed rows

Development of a majorization-based decreasing criterion for row-sum vectors

Abstract

A combinatorial rectangle may be viewed as a matrix whose entries are all +-1. The discrepancy of an m by n matrix is the maximum among the absolute values of its m row sums and n column sums. In this paper, we investigate combinatorial rectangles with minimum discrepancy (0 or 1 for each line depending on the parity). Specifically, we get explicit formula for the number of matrices with minimum L^infinity-discrepancy up to 4 rows, and establish the order of magnitude of the number of such matrices with m rows and n columns while m is fixed and n approaches infinity. By considering the number of column-good matrices with a fixed row-sum vector, we have developed a theory of decreasing criterion on based row-sum vectors with majorization relation, which turns out to be a helpful tool in the proof of our main theorems.