Resonance between Cantor Sets

Let Ca 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(Ca + Cb) = min(dim(Ca) + dim(Cb), 1), where dim is Hausdorff dimension. More generally, given two self-similar sets K,K′ in R and a scaling parameter s > 0, if the dimension of the arithmetic sum K + sK′ is strictly smaller than dim(K) + dim(K′) ≤ 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.