Convolutions of Cantor Measures Without Resonance

Denote by \({\mu }_a\) the distribution of the random sum \((1-a){\sum }_{\infty }^{j=0}{\omega }_ja_j\), 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 \({\mu }_a\ast ({\mu }_b\circ S_{-1}^{\lambda })\), where \(S_{\lambda }(x)=\lambda x\) is a rescaling map. We prove that if the ratio \(logb/loga\) is irrational and \(\lambda \neq 0\), 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 \(\lambda\) the convolution \({\mu }_{1/4}\ast ({\mu }_{1/3}\circ S_{-1}^{\lambda })\) 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.

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