On the probability that a random $\pm 1$-matrix is singular

We report some progress on the old problem of estimating the probability, ${P_n}$, that a random $n \times n \pm 1$-matrix is singular:

Theorem. There is a positive constant $\varepsilon$ for which ${P_n} < {(1 - \varepsilon )^n}$.

This is a considerable improvement on the best previous bound, ${P_n} = O(1/\sqrt n )$, given by Komlós in 1977, but still falls short of the often-conjectured asymptotical formula ${P_n} = (1 + o(1)){n^2}{2^{1 - n}}$.

The proof combines ideas from combinatorial number theory, Fourier analysis and combinatorics, and some probabilistic constructions. A key ingredient, based on a Fourier-analytic idea of Halász, is an inequality (Theorem 2) relating the probability that $\underline a \in {{\mathbf{R}}^n}$ is orthogonal to a random $\underline \varepsilon \in {\{ \pm 1\} ^n}$ to the corresponding probability when $\underline \varepsilon$ is random from ${\{ - 1,0,1\} ^n}$ with $Pr({\varepsilon _i} = - 1) = Pr({\varepsilon _i} = 1) = p$ and ${\varepsilon _i}$'s chosen independently.