A classical theorem (Marstrand 1954) asserts that, given any Borel subset , the obvious inequality of the Hausdorff dimensions: is in fact an equality for almost all orthogonal projections . As is often the case it is usually very dificult to prove equality for a given projection.
Preliminary description: Michael HOCHMAN and Pablo SCHMERKIN have posted a preprint presenting a new method for doing this. They actually deal with the stronger statement involving the dimension of measures. The key step is a lower semicontinuity property under an assumption of regularity (the process defined by zooming at typical points must be stationary). The classical a.e. result then yields an open and dense set, which can be controlled using invariance of the considered measures under large groups.