Title: Projection Theorems Using Effective Dimension,
Abstract: In this paper we use the theory of computing to study fractal dimensions of
projections in Euclidean spaces. A fundamental result in fractal geometry is
Marstrand's projection theorem, which shows that for every analytic set E, for
almost every line L, the Hausdorff dimension of the orthogonal projection of E
onto L is maximal. We use Kolmogorov complexity to give two new results on the
Hausdorff and packing dimensions of orthogonal projections onto lines. The
first shows that the conclusion of Marstrand's theorem holds whenever the
Hausdorff and packing dimensions agree on the set E, even if E is not analytic.
Our second result gives a lower bound on the packing dimension of projections
of arbitrary sets. Finally, we give a new proof of Marstrand's theorem using
the theory of computing.