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Journal Article

The Local Dimensions of the Bernoulli Convolution Associated with the Golden Number

Tian-You Hu
Transactions of the American Mathematical Society
Vol. 349, No. 7 (Jul., 1997), pp. 2917-2940
Stable URL: http://www.jstor.org/stable/2155559
Page Count: 24

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Abstract

Let X1, X2, ... 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 - ρ - ⋯ - ρs = 0, let μ be the probability measure induced by S = ∑∞ i = 1 ρiX i. For any x in the range of S, let $d(\mu, x) = \lim_{r\rightarrow 0^+} \log \mu(\lbrack x - r, x + r\rbrack)/\log r$ be the local dimension of μ at x whenever the limit exists. We prove that α* = -log 2/log ρ and α* = -log δ/s log ρ - log 2/log ρ, where $\delta = (\sqrt 5 - 1)/2$, are respectively the maximum and minimum values of the local dimensions. If s = 2, then ρ is the golden number, and the approximate numerical values are α* ≈ 1.4404 and α* ≈ 0.9404.

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