  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.
