Dimension and projections in normed spaces and Riemannian manifolds

Abstract

This thesis is concerned with the behavior of Hausdorff measure and Hausdorff dimension under projections. In 1954, Marstrand proved that given a Borel set A Ϲ R² of dimension strictly larger than 1, for almost every line L that passes through the origin, the orthogonal projection of A onto L is a set of positive Hausdorff 1-measure. This theorem marked the start of a long sequence of results in the same spirit that are nowadays known as Marstrand-type projection theorems. In the first part of this thesis, we establish Marstrand-type projection theorems for projections induced by linear foliations as well as for closest-point projections onto hyperplanes in finite dimensional normed spaces. By the same methods we obtain a Besicovitch-Federer-type characterization of purely unrectifiable sets in terms of these families of projections. Moreover, we give an example underlining the sharpness of our results. In the second part of the thesis, we establish Marstrand-type as well as Besicovitch-Federer-type projection theorems for orthogonal projections along geodesics in hyperbolic space as well as in the two-sphere. Several of these results are achievable by two different methods: potential theoretic methods and Fourier analytic methods. We discuss the scope of each of these methods in both settings.