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:
► Sixty Years of Fractal Projections (with K. J. Falconer & X. Jin), arXiv
Fractal Geometry and Stochastics V, Birkhäuser, Progress in Probability, 2015, (Eds. C. Bandt, K. J. Falconer & M. Zähle).
Some of my work on the dimension theory of projections includes:
► Assouad dimension influences the box and packing dimensions of orthogonal projections (with K. J. Falconer & P. Shmerkin), arXiv
Journal of Fractal Geometry (to appear).
► Projection theorems for intermediate dimensions (S. A. Burrell & K. J. Falconer), arXiv
Journal of Fractal Geometry (to appear).
► Scaling scenery of (×m,×n) invariant measures (with A. Ferguson & T. Sahlsten), arXiv
Advances in Mathematics, 268, (2015), 564-602.
The Assouad dimension behaves rather differently to the Hausdorff and box dimension in the context of projections. Work in this direction includes:
► A nonlinear projection theorem for Assouad dimension and applications, arXiv
Journal of the London Mathematical Society (to appear).
► Attainable values for the Assouad dimension of projections (with A. Käenmäki), arXiv
Proceedings of the American Mathematical Society, 148, (2020), 3393-3405.
► Distance sets, orthogonal projections, and passing to weak tangents, arXiv
Israel Journal of Mathematics, 226, (2018), 851-875.
► The Assouad dimensions of projections of planar sets (with T. Orponen), arXiv
Proceedings of the London Mathematical Society, 114, (2017), 374-398.