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Mathematics  2007

# Resonance between Cantor sets

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Abstract:

Let \$C_a\$ be the central Cantor set obtained by removing a central interval of length \$1-2a\$ from the unit interval, and continuing this process inductively on each of the remaining two intervals. We prove that if \$\log b/\log a\$ is irrational, then \[ \dim(C_a+C_b) = \min(\dim(C_a) + \dim(C_b),1), \] where \$\dim\$ is Hausdorff dimension. More generally, given two self-similar sets \$K,K'\$ in \$\RR\$ and a scaling parameter \$s>0\$, if the dimension of the arithmetic sum \$K+sK'\$ is strictly smaller than \$\dim(K)+\dim(K') \le 1\$ (``geometric resonance''), then there exists \$r<1\$ such that all contraction ratios of the similitudes defining \$K\$ and \$K'\$ are powers of \$r\$ (``algebraic resonance''). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation.

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