Thin circulant matrices and lower bounds on the complexity of some Boolean operators

We prove a lower bound $\Omega\left(\frac{k+l}{k^2l^2}N^{2-\frac{k+l+2}{kl}}\right)$ on the maximal possible weight of a $(k,l)$-free (that is, free of all-ones $k\times l$ submatrices) Boolean circulant $N \times N$ matrix. The bound is close to the known bound for the class of all $(k,l)$-free matrices. As a consequence, we obtain new bounds for several complexity measures of Boolean sums’ systems and a lower bound $\Omega(N^2\log^{-6} N)$ on the monotone complexity of the Boolean convolution of order $N$.

💡 Research Summary

The paper investigates the maximal possible weight (i.e., the number of ones) of Boolean circulant N × N matrices that avoid an all‑ones k × l submatrix, a property known as (k,l)‑free or “thin”. While the classic Zarankiewicz problem gives a lower bound for arbitrary (k,l)‑free matrices, no comparable bound was known for the much more constrained class of circulant matrices, whose entire structure is determined by a single row. The authors prove a new lower bound \