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

Abstract: We show that for a self-similar set \(A\) in \(\mathbb{R}^d\) for \(d \geq 3\), if its rotation components of \(A\) generate a dense subgroup of \(\mathrm{SO}(d)\) 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 \(A \cap \pi^{-1}_V(y)\) has Hausdorff dimension \(\dim_H A -k\). 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.