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Convolutions of Cantor measures without resonance

Shmerkin, PS, Nazarov, F and Peres, Y (2011) Convolutions of Cantor measures without resonance Israel Journal of Mathematics, 187 (1). 93 - 116. ISSN 0021-2172

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Abstract

Denote by $\mu_a$ the distribution of the random sum $ \; \; (1-a) \sum_{j=0}^\infty \omega_j a^j$, where $\mathbf{P}(\omega_j=0)=\mathbf{P}(\omega_j=1)=1/2$ and all the choices are independent. For $0<a<1/2$, the measure $\mu_a$ is supported on $C_a$, the central Cantor set obtained by starting with the closed united interval, removing an open central interval of length $(1-2a)$, and iterating this process inductively on each of the remaining intervals. We investigate the convolutions $\mu_a * (\mu_b \circ S_\lambda^{-1})$, where $S_\lambda(x)=\lambda x$ is a rescaling map. We prove that if the ratio $\log b/\log a$ is irrational and $\lambda\neq 0$, then \[ D(\mu_a *(\mu_b\circ S_\lambda^{-1})) = \min(\dim_H(C_a)+\dim_H(C_b),1), \] where $D$ denotes any of correlation, Hausdorff or packing dimension of a measure. We also show that, perhaps surprisingly, for uncountably many values of $\lambda$ the convolution $\mu_{1/4} *(\mu_{1/3}\circ S_\lambda^{-1})$ is a singular measure, although $\dim_H(C_{1/4})+\dim_H(C_{1/3})>1$ and $\log (1/3) /\log (1/4)$ is irrational.

Item Type: Article
Additional Information: The original publication is available at http://www.springerlink.com
Divisions: Faculty of Engineering and Physical Sciences > Mathematics
Depositing User: Symplectic Elements
Date Deposited: 27 Jan 2012 09:51
Last Modified: 09 Jun 2014 13:21
URI: http://epubs.surrey.ac.uk/id/eprint/121088

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