Matrix Rigidity and the Croot-Lev-Pach Lemma

Matrix rigidity is a notion put forth by Valiant (1977) as a means for proving arithmetic circuit lower bounds. A matrix is rigid if it is far, in Hamming distance, from any low-rank matrix. Despite decades of effort, no explicit matrix rigid enough to carry out Valiant's plan has been found. Recently, Alman and Williams (STOC'17) showed that, contrary to common belief, the Walsh-Hadamard matrices cannot be used for Valiant's program as they are not sufficiently rigid. Our main result is a similar non-rigidity theorem for any q(n) x q(n) matrix M of the form M(x,y) = f (x y), where f : F-q(n) -> F-q is any function and F-q is a fixed finite field of q elements (n goes to infinity). The theorem follows almost immediately from a recent lemma of Croot, Lev and Pach (2017) which is also the main ingredient in the recent solution of the famous cap-set problem by Ellenberg and Gijswijt (2017).