# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## The local dimensions of the Bernoulli convolution associated with the golden numberHTML articles powered by AMS MathViewer

by Tian-You Hu
Trans. Amer. Math. Soc. 349 (1997), 2917-2940 Request permission

## Abstract:

Let $X_1,X_2,\dotsc$ be a sequence of i.i.d. random variables each taking values of 1 and $-1$ with equal probability. For $1/2<\rho <1$ satisfying the equation $1-\rho -\dotsb -\rho ^s=0$, let $\mu$ be the probability measure induced by $S=\sum _{i=1}^\infty \rho ^iX_i$. For any $x$ in the range of $S$, let $d(\mu ,x)=\lim _{r\to 0^+}\log \mu ([x-r,x+r])/\log r$ be the local dimension of $\mu$ at $x$ whenever the limit exists. We prove that $\alpha ^*=-\frac {\log 2}{\log \rho }\quad \text {and}\quad \alpha _*=-\frac {\log \delta }{s\log \rho }-\frac {\log 2}{\log \rho },$ where $\delta =(\sqrt {5}-1)/2$, are respectively the maximum and minimum values of the local dimensions. If $s=2$, then $\rho$ is the golden number, and the approximate numerical values are $\alpha ^*\approx 1.4404$ and $\alpha _*\approx 0.9404$.
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