Convolutions of Cantor Measures Without Resonance

Denote by μa the distribution of the random sum (1−a)∑∞j=0ωjaj, where P(ωj=0)=P(ωj=1)=1/2 and all the choices are independent. For 0<a<1/2, the measure μa is supported on Ca, 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 μa∗(μb∘S−1λ), where Sλ(x)=λx is a rescaling map. We prove that if the ratio logb/loga is irrational and λ≠0, then

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 μ1/4∗(μ1/3∘S−1λ) is a singular measure, although dimH(C1/4)+dimH(C1/3)>1 and log(1/3)/log(1/4) is irrational.