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

2024/12/08 by Bárány, Balázs, Rams, Michał · 1 citation
#28A80 60G30 60G57 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2412.06008

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 Hölder 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.

Cited by

• On the Fourier transform of random Bernoulli convolutions

Related

• Self-similar sets and measures on the line
• The interior of randomly perturbed self-similar sets on the line
• Smooth projections of self-similar measures
• Absolute continuity of self-similar measures on the plane
• On self-similar measures with absolutely continuous projections and dimension conservation in each direction