On the dimension of Bernoulli convolutions for all transcendental parameters

By P. P. Varj'u. Annals of Mathematics, 2018

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The Bernoulli convolution $\nu_\lambda$ with parameter $\lambda\in(0,1)$ is the probability measure supported on $\mathbf{R}$ that is the law of the random variable $\sum\pm\lambda^n$, where the $\pm$ are independent fair coin-tosses. We prove that $\dim\nu_\lambda=1$ for all transcendental $\lambda\in(1/2,1)$.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ee7a65dd16f7b0922b0e2c901902a0f14865b92e