Resonance between Cantor sets

Abstract Let C a be the central Cantor set obtained by removing a central interval of length 1−2 a 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 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.