The sharp Hausdorff measure condition for length of projections

In a recent paper, Pertti Mattila asked which gauge functions $ϕ$ have the property that for any planar Borel set $A$ with positive Hausdorff measure in gauge $ϕ$, the projection of $A$ to almost every line has positive length. We show that integrability near zero of $ϕ(r)/(r^2)$, which is known to be sufficient for this property, is also necessary if $ϕ$ is regularly varying. Our proof is based on a random construction adapted to the gauge function.

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