On $L^p$ Density of Self-Similar Measures and Continuous Bernoulli Convolutions

I want to share discoveries made by GPT-6 Astra last week, providing an elegant new idea for questions I wondered about for years. All new theorems stated in this talk are fully formalised in Lean.

A self-similar measure is a stationary measure from appying random contracting similarities on Rd. They are central objects in fractal geometry. Understanding when they are absolutely continuous, that is that they have a density with respect to the Lebesgue measure, is intricate. I present the new result that the density of an absolutely continuous self-similar measure is always in Lp for some p greater than 1. The short proof of this result results in a quantitative p in terms of the mass that the self-similar measure gives to sets of small Lebesgue measure.

The most studied example of self-similar measures are Bernoulli convolutions. We apply the new insight on Lp densities to give explicit examples of Bernoulli convolutions with a continuous density, which was previously only know for paramaters of Mahler measure exactly 2. For example, we show that the Bernoulli convolution of parameter 1-1/n is has a continuous density for all n larger than 588. The latter result was previously not known for any n, while it was only known that they are absolutely continuous, that is have a density in L1, for n bigger than about 1020.