Convolutions of cantor measures without resonance.

Denote by µ the distribution of the random sum $$(1 - a)\sum\nolimits_{j = 0}^\infty {{w_j}{a^j}} $$, where P( ω = 0) = P( ω = 1) = 1/2 and all the choices are independent. For 0 < a < 1/2, the measure µ is supported on C, the central Cantor set obtained by starting with the closed united interval, removing an open central interval of length (1 − 2 a), and iterating this process inductively on each of the remaining intervals. We investigate the convolutions µ* (µ ∘ S), where S( x) = λx is a rescaling map. We prove that if the ratio log b/ log a 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 µ* (µ ∘ S) is a singular measure, although dim( C) + dim( C) > 1 and log(1/3)/ log(1/4) is irrational. [ABSTRACT FROM AUTHOR]