We consider a one parameter family of self-similar sets of overlapping construction. We study the exceptional set; that is the set of those parameters for which the correlation dimension is smaller than the similarity dimension. We find a connection between the exceptional set and the multifractal analysis of a measure.
"Multifractals and the Dimension of Exceptions." Real Anal. Exchange 27 (1) 191 - 208, 2001/2002.