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

DOI:
https://doi.org/10.1090/S0002-9947-97-01474-8

MathSciNet review:
1321578

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Abstract | References | Similar Articles | Additional Information

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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Additional Information

**Tian-You Hu**

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

Email:
HUT@gbms01.uwgb.edu

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