On the absolute continuity of radial and linear projections of missing digits measures

Abstract

In this paper, we study the absolute continuity of radial projections of missing digits measures. We show that for large enough missing digits measures $\lambda$ on ${\mathbb{R}}^n,n\ge 2,$ for all $x\in {\mathbb{R}}^n\setminus \mathrm{supp}(\lambda ),$ ${\Pi }_x(\lambda )$ is absolutely continuous with a density function in $L^2(S^{n-1}).$ Our method applies to linear projections as well. In particular, we show that for $\lambda$ as above, the linearly projected measure $P_{\theta }(\lambda )$ is absolutely continuous with a continuous density function for almost all directions $\theta \in S^{n-1}.$ This implies a version of Palis' conjecture for missing digits sets.