Katrin Fässler

On restricted families of projections in ℝ3

We study projections onto non-degenerate one-dimensional families of lines and planes in R 3 . Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most 12 -dimensional sets [Math Processing Error] is typically preserved under one-dimensional families of projections onto lines. We improve the result by an e , proving that if [Math Processing Error], then the packing dimension of the projections is almost surely at least [Math Processing Error]. For projections onto planes, we obtain a similar bound, with the threshold 12 replaced by 1 . In the special case of self-similar sets [Math Processing Error] without rotations, we obtain a full Marstrand-type projection theorem for 1-parameter families of projections onto lines. The [Math Processing Error] case of the result follows from recent work of M. Hochman, but the [Math Processing Error] part is new: with this assumption, we prove that the projections have positive length almost surely.

Constancy results for special families of projections

Let { = V × ℝ l : V ∈ G(n−l,m−l )} be the family of m -dimensional subspaces of ℝ n containing {0} × ℝ l , and let : ℝ n → be the orthogonal projection onto . We prove that the mapping V ↦ Dim ( B ) is almost surely constant for any analytic set B ⊂ ℝ n , where Dim denotes either Hausdorff or packing dimension.