Smoothness of random self-similar measures on the line and the existence of interior points

TL;DR

This paper investigates the conditions under which random self-similar measures on the line have smooth densities and contain interior points, revealing that higher similarity dimensions lead to absolute continuity and interior points.

Contribution

It establishes that measures with symbolic local dimension greater than 1 are almost surely absolutely continuous with Hölder continuous densities, and that sets with dimension greater than 1 contain interior points.

Findings

Measures with local dimension > 1 are absolutely continuous almost surely.

Random self-similar sets with dimension > 1 contain interior points.

Density functions are Hölder continuous under certain conditions.

Abstract

In this paper, we study the smoothness of the density function of absolutely continuous measures supported on random self-similar sets on the line. We show that the natural projection of a measure with symbolic local dimension greater than 1 at every point is absolutely continuous with H\"older continuous density almost surely. In particular, if the similarity dimension is greater than 1 then the random self-similar set on the line contains an interior point almost surely.