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

TL;DR

This paper proves that radial and linear projections of missing digits measures in higher dimensions are absolutely continuous, with the densities having specific regularity properties, confirming a version of Palis' conjecture.

Contribution

It establishes absolute continuity and regularity of projections of missing digits measures, extending results to both radial and linear projections in higher dimensions.

Findings

Radial projections are absolutely continuous with $L^2$ densities.

Linear projections are absolutely continuous with continuous densities for almost all directions.

Results confirm a version of Palis' conjecture for missing digits sets.

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 $\mathbb{R}^n,n\geq 2,$ 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.