Convolutions of Cantor measures without resonance

TL;DR

This paper studies convolutions of Cantor measures generated by random sums, revealing conditions under which their dimensions add up to a maximum of 1 or result in singular measures, especially when the ratio of their parameters is irrational.

Contribution

It proves that for irrational log ratios, the convolution's dimension equals the minimum of the sum of individual dimensions and 1, and identifies cases where the convolution is singular.

Findings

Convolution dimension equals min of sum of individual dimensions and 1 for irrational log ratios.

Existence of uncountably many λ where the convolution measure is singular.

Convolution measures can be singular despite their dimensions summing to more than 1.

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.