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.