On the singularity probability of random Bernoulli matrices

Let $n$ be a large integer and let $M_n$ be a random $n \times n$ 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, Komlós and Szemerédi, 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.