We prove that finiteness of the integral of $\phi(r)/r^2$ at 0 is a sharp condition on the gauge function $\phi$ which ensures that a planar Borel set with positive Hausdorff measure in the gauge $\phi$ has projection of positive length to almost every line. Sufficiency of this condition is known; we establish its sharpness via a random construction. This is joint work with Yuval Peres.