Multifractal analysis for convolutions of overlapping Cantor measures

Unlike the case for self-similar measures satisfying the open set condition, it has been shown that the m-fold convolution of the uniform Cantor measure on the classical middle-third Cantor set has isolated points in its multifractal spectrum for any m ≥ 3. We show that this phenomena of isolated points holds for quite general Cantor measures on Cantor sets that can be far from self-similar. We also prove, in contrast, that if the convolution is understood on the group [0, 1], rather than on R{double-struck}, then the multifractal spectrum of the 3-fold convolution of the uniform Cantor measure is an interval.