Convolutions of Cantor measures without resonance

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 ${\mu }_a\ast ({\mu }_b\circ S_{\lambda }^{-1})$, where $S_{\lambda }(x)=\lambda x$ is a rescaling map. We prove that if the ratio $\log b/\log a$ is irrational and $\lambda \ne 0$, then \[ D(μ_a *(μ_b\circ S_λ^{-1})) = \min(\dim_H(C_a)+\dim_H(C_b),1), \] 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 ${\mu }_{1/4}\ast ({\mu }_{1/3}\circ S_{\lambda }^{-1})$ 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.

References in corpus (1)

• On the arithmetic sums of Cantor sets