Convolutions of Cantor measures without resonance

Nazarov, Fedor and Peres, Yuval and Shmerkin, Pablo (2009) Convolutions of Cantor measures without resonance. [MIMS Preprint]

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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 $P(\omega_j=0)=P(\omega_j=1)=1/2$ 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: MIMS Preprint
Uncontrolled Keywords: CICADA, correlation dimension, convolutions, self-similar measures, resonance
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 28 Measure and integration
MSC 2010, the AMS's Mathematics Subject Classification > 42 Fourier analysis
Depositing User: Mr Pablo Shmerkin
Date Deposited: 14 Oct 2009
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/1333

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