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 γ₂ factorization norm at most $1$. Building on recent work by Balla, Hambardzumyan, and Tomon, we show that for any boolean matrix with γ₂ norm at most $λ$, there exists a a collection of row- and column-disjoint 1-monochromatic submatrices that together cover a significant portion (at least a 1/2^2O(λ) fraction) of its $1$-entries.