## 2009.71: Convolutions of Cantor measures without resonance

2009.71:
Fedor Nazarov, Yuval Peres and Pablo Shmerkin
(2009)
*Convolutions of Cantor measures without resonance.*

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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 |
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Uncontrolled Keywords: | CICADA, correlation dimension, convolutions, self-similar measures, resonance |

Subjects: | MSC 2000 > 28 Measure and integration MSC 2000 > 42 Fourier analysis |

MIMS number: | 2009.71 |

Deposited By: | Mr Pablo Shmerkin |

Deposited On: | 14 October 2009 |

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