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

Bal\'azs B\'ar\'any; Micha{\l} Rams

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

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arxiv 2412.06008 v2 pith:PY6ORCG2 submitted 2024-12-08 math.DS math.PR

Balázs Bárány , Micha{l} Rams This is my paper · ORCID

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keywords continuouslinerandomself-similarabsolutelyalmostdensitydimension

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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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On the Fourier transform of random Bernoulli convolutions
math.DS 2025-07 accept novelty 7.0 of 10

For random Bernoulli convolutions, the Fourier transform is in L^1 almost surely whenever λ_g > 2/π, giving absolute continuity and non-empty interior; polynomial Fourier decay holds for every λ_g.

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