Projections of fractals

One of the most well-studied and fundamental questions in fractal geometry is: how do fractals behave under projection? (e.g. orthogonal projection of the plane onto lines). Seminal work of Marstrand, Mattila and Kaufmann proved that the Hausdorff dimension of the projection of a Borel set in d-dimensional space onto a k-dimensional subspace is almost surely given by the minimum of the Hausdorff dimension of the set and k. This result, and several variants, have led to an enormous amount of research in the literature. Together with Kenneth Falconer and Xiong Jin, I wrote a survey on the topic: Some of my work on the dimension theory of projections includes: The Assouad dimension behaves rather differently to the Hausdorff and box dimension in the context of projections. Work in this direction includes: