  In graph theory, the Szemer\'edi regularity lemma gives a decomposition of
the indicator function for any graph $G$ into a structured component, a uniform
part, and a small error. This result, in conjunction with a counting lemma that
guarantees many copies of a subgraph $H$ provided a copy of $H$ appears in the
structured component, is used in many applications to extremal problems. An
analogous decomposition theorem exists for functions over $\mathbb{F}_p^n$.
Specializing to $p=2$, we obtain a statement about the indicator functions of
simple binary matroids. In this paper we extend previous results to prove a
corresponding counting lemma for binary matroids. We then apply this counting
lemma to give simple proofs of some known extremal results, analogous to the
proofs of their graph-theoretic counterparts, and discuss how to use similar
methods to attack a problem concerning the critical numbers of dense binary
matroids avoiding a fixed submatroid.
