Cai-Yun Ma: Box and Packing Dimensions of Orthogonal Projections of Homogeneous Gatzouras–Lalley Carpets and Phase Transitions

The study of orthogonal projections of sets and measures is an important topic in geometric measure theory. About two decades ago, Falconer and Howroyd established the projection theorems for the packing, and upper and lower box dimensions. They showed that for every Borel set 𝐴 in ℝ^𝑑, each of the packing, upper box and lower box dimensions of the projection 𝑃𝑉⁡(𝐴) of 𝐴 takes a constant value, which can be expressed as a certain dimension profile, for almost all 𝑘-dimensional subspaces 𝑉. However, these dimension profiles are defined indirectly and very difficult to compute. Recently, we have succeeded in obtaining the precise values of these dimension profiles for homogeneous Gatzouras-Lalley carpets, which exhibit remarkable phase transitions. The talk is based on joint work with Dejun Feng and Károly Simon.

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