Convolutions of Cantor measures without resonance - Mathematics > Classical Analysis and ODEsReport as inadecuate




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Abstract: Denote by $\mu a$ the distribution of the random sum $1-a \sum {j=0}^\infty\omega j a^j$, where $P\omega j=0=P\omega j=1=1-2$ and all the choices areindependent.
For $0We prove that if the ratio $\logb-\log a$ is irrational and $\lambda eq 0$, then \ D\mu a *\mu b\circS \lambda^{-1} = \min\dim HC a+\dim HC b,1, \ where $D$ denotes any ofcorrelation, Hausdorff or packing dimension of a measure.We also show that, perhaps surprisingly, for uncountably many values of$\lambda$ the convolution $\mu {1-4} *\mu {1-3}\circ S \lambda^{-1}$ is asingular measure, although $\dim HC {1-4}+\dim HC {1-3}>1$ and $\log 1-3-\log 1-4$ is irrational.



Author: Fedor Nazarov, Yuval Peres, Pablo Shmerkin

Source: https://arxiv.org/



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