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arXiv · 2412.06008

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

Balázs Bárány·Michał Rams
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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.

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Balázs Bárány, Michał Rams. 2024-12-08. Smoothness of random self-similar measures on the line and the existence of interior points. https://arxiv.org/abs/2412.06008

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