Ergodic Theory and Dynamical Systems

Research Article

Resonance between Cantor sets


a1 Microsoft Research, Redmond and Departments of Statistics and Mathematics, University of California, Berkeley, USA (email:

a2 Departments of Mathematics and Statistics, University of Jyväskylä, Finland (email:


Let Ca 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 xs211D 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.

(Received May 18 2007)

(Revised March 28 2008)