Hyperarithmetic directions can all be exceptional for Marstrand's projection theorem

Abstract

We show that there is a ${\Pi }_2^0$ subset $B$ of the Euclidean plane, of Hausdorff dimension 1, such that for every line $\ell$ through the origin with hyperarithmetic direction, the projection of $B$ onto $\ell$ has Hausdorff dimension 0, thus is exceptional for Marstrand's projection theorem. It follows that for any computable ordinal $\alpha$, being $\alpha$-random does not guarantee the ``almost-all'' statement of the projection theorem.