ZFC counterexamples to Marstrand’s projection theorem

Establish whether a counterexample to conclusion (1) of Marstrand’s Projection Theorem, namely a set whose projections fail to have the expected Hausdorff dimension for almost every direction, can be constructed within ZFC without additional set-theoretic assumptions.

Background

Marstrand’s Projection Theorem states that if B is an analytic subset of the Euclidean plane, then for almost every line through the origin, the projection of B onto that line has Hausdorff dimension equal to the minimum of 1 and the Hausdorff dimension of B. The introduction explains that Davies constructed a set violating this conclusion under the Continuum Hypothesis, while stronger assumptions such as V=L were used to obtain co-analytic counterexamples.

The unresolved issue is whether any counterexample of this Davies type can be produced in the ordinary axiomatic framework of ZFC itself. This problem concerns the precise set-theoretic strength required to violate the almost-everywhere projection conclusion when the analyticity assumption is weakened or removed.