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 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 $K,K^{\mathrm{\prime }}$ in $\text{RR}$ and a scaling parameter $s>0$, if the dimension of the arithmetic sum $K+s{K}^{\mathrm{\prime }}$ is strictly smaller than $\dim (K)+\dim (K^{\mathrm{\prime }})\le 1$ (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.