Characterising the big pieces of Lipschitz graphs property using projections

We characterise the big pieces of Lipschitz graphs property in terms of projections. Roughly speaking, we prove that if a large subset of an $n$-Ahlfors-David regular set $E\subset {\mathbb{R}}^d$ has plenty of projections in $L^2$, then a large part of $E$ is contained in a single Lipschitz graph. This is closely related to a question of G. David and S. Semmes.