Dimension-free discrepancy for bounded max-norm Boolean matrices

Establish that for every positive real number gamma there exists a constant k_gamma such that every Boolean matrix M with gamma_2(M) at most gamma has discrepancy at most k_gamma.

Background

The discrepancy of a Boolean matrix is the minimum, over sign colorings of its columns, of the maximum absolute row sum; it measures the largest imbalance induced by a two-coloring. The paper relates discrepancy to gamma_2 through factorization norms and proves a discrepancy bound for matrices admitting sparse, bounded-entry factorizations.

A positive solution to the bounded blocky-decomposition conjecture would imply this discrepancy conjecture, because every integer matrix M satisfies disc(M) at most 2 block(M)2. The paper does not resolve whether bounded gamma_2 alone forces dimension-free discrepancy.

References

Conjecture. For every $\gamma>0$ there exists $k_{\gamma}$ such that the following holds. Let $M$ be a Boolean matrix such that $\gamma_2(M)\leq \gamma$. Then $\disc(M)\leq k_{\gamma}$.

— Factorization norms and Zarankiewicz problems  (2502.18429 - Tomon, 25 Feb 2025) in Section 1, subsection "Discrepancy theory," Conjecture 2