Block structure in boolean matrices of bounded factorization norm

A boolean matrix is blocky if its $1$-entries form a collection of 1-monochromatic submatrices that are disjoint in both rows and columns. Blocky matrices are precisely the set of boolean matrices with $\gamma_2$ factorization norm at most $1$. Building on recent work by Balla, Hambardzumyan, and Tomon, we show that for any boolean matrix with $\gamma_2$ norm at most $\lambda$, there exists a a collection of row- and column-disjoint 1-monochromatic submatrices that together cover a significant portion (at least a $1/2^{2^{O(\lambda)}}$ fraction) of its $1$-entries.