The sharp Hausdorff measure condition for length of projections

Abstract:

In a recent paper, Pertti Mattila asked which gauge functions $\varphi$ have the property that for any Borel set $A\subset \mathbb {R}^2$ with Hausdorff measure $\mathcal {H}^\varphi (A)>0$, the projection of $A$ to almost every line has positive length. We show that finiteness of $\int _0^1\frac {\varphi (r)}{r^2} dr$, which is known to be sufficient for this property, is also necessary for regularly varying $\varphi$. Our proof is based on a random construction adapted to the gauge function.