  Matrix rigidity is a notion put forth by Valiant 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 efforts, no explicit
matrix rigid enough to carry out Valiant's plan has been found. Recently, Alman
and Williams showed, contrary to common belief, that the $2^n \times 2^n$
Hadamard matrix could not be used for Valiant's program as it is not
sufficiently rigid. In this note we observe a similar `non rigidity' phenomena
for any $q^n \times q^n$ matrix $M$ of the form $M(x,y) = f(x+y)$, where
$f:F_q^n \to F_q$ 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 which is also the main ingredient in the
recent solution of the cap-set problem.
