Resonance between Cantor sets

Let C(a) be the central Cantor set obtained by removing a central interval of length 1 - 2a from the unit interval, and then 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 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 oil the projections of planar self-similar sets generated by ail iterated function system that includes a scaled irrational rotation.