  Let $n$ be a large integer and $M_n$ be a random $n$ by $n$ matrix whose
entries are i.i.d. Bernoulli random variables (each entry is $\pm 1$ with
probability 1/2). We show that the probability that $M_n$ is singular is at
most $(3/4 +o(1))^n$, improving an earlier estimate of Kahn, Koml\'os and
Szemer\'edi, as well as earlier work by the authors. The key new ingredient is
the applications of Freiman type inverse theorems and other tools from additive
combinatorics.

Combinatorics