If the symbolic local dimension is >1 and the perturbation has Fourier decay, the random self-similar measure has an almost-surely Hölder continuous density, implying the random fractal contains an interior point almost surely.
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Smoothness of random self-similar measures on the line and the existence of interior points
If the symbolic local dimension is >1 and the perturbation has Fourier decay, the random self-similar measure has an almost-surely Hölder continuous density, implying the random fractal contains an interior point almost surely.