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Transactions of the American Mathematical Society

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The local dimensions of the
Bernoulli convolution associated with
the golden number

Author: Tian-You Hu
Journal: Trans. Amer. Math. Soc. 349 (1997), 2917-2940
MSC (1991): Primary 28A80; Secondary 42A85
MathSciNet review: 1321578
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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

\begin{displaymath}d(\mu ,x)=\lim _{r\to 0^+}\log \mu([x-r,x+r])/\log r\end{displaymath}

be the local dimension of $\mu $ at $x$ whenever the limit exists. We prove that

\begin{displaymath}\alpha ^*=-\frac {\log 2}{\log \rho}\quad \text{and}\quad \alpha _*=-\frac {\log \delta }{s\log \rho}-\frac {\log 2}{\log \rho},\end{displaymath}

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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Additional Information

Tian-You Hu
Affiliation: Department of Mathematics, University of Wisconsin-Green Bay, Green Bay, Wisconsin 54311

Keywords: Bernoulli convolution, Fibonacci sequence, local dimension, PV-number
Received by editor(s): August 23, 1994
Received by editor(s) in revised form: January 25, 1995
Article copyright: © Copyright 1997 American Mathematical Society