  We show that, for each real number $\alpha > 0$ and odd integer $k\ge 5$
there is an integer $c$ such that, if $M$ is a simple binary matroid with $|M|
\ge \alpha 2^{r(M)}$ and with no $k$-element circuit, then $M$ has critical
number at most $c$. The result is an easy application of a regularity lemma for
finite abelian groups due to Green.
