Paper detail

Convolutions of Cantor measures without resonance

Denote by $μ_a$ the distribution of the random sum $(1-a) \sum_{j=0}^\infty ω_j a^j$, where $P(ω_j=0)=P(ω_j=1)=1/2$ and all the choices are independent. For $0<a<1/2$, the measure $μ_a$ 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 $μ_a * (μ_b \circ S_λ^{-1})$, where $S_λ(x)=λx$ is a rescaling map. We prove that if the ratio $\log b/\log a$ is irrational and $λ\neq 0$, 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 $λ$ the convolution $μ_{1/4} *(μ_{1/3}\circ S_λ^{-1})$ 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.