Xiong Jin: Absolute continuity of projections and dimension conservation for self-similar sets and measures

Speaker
Xiong Jin, University of Manchester

Abstract
We show that for a self-similar set \(A\) in \(\mathbb{R}^d\) for \(d\ge 3\), if its rotation components of \(A\) generate a dense subgroup of SO(d) and its Hausdorff dimension satisfies \(\dim_H A> k\) for some integer \(1\leq k \<d\), then its orthogonal projection \(\pi_V(A)\) to every \(k\)-dimensional linear subspace \(V\) has positive \(k\)-Lebesgue measure, and for \(k\)-Lebesgue almost every point \(y \in \pi_V(A)\), its fibre \(A\cap \pi_V^{-1}(y)\) has Hausdorff dimension \(\dim_HA-k\). The proof uses the \(L^1\)-increments of the Gaussian smoothing of the projected self-similar measures and Varju’s weak gap theorem for random walks on compact semisimple Lie groups. This is a joint work with Tuomas Sahlsten.