Ergodic Theory and Dynamical Systems

Research Article

Resonance between Cantor sets

YUVAL PERESa1 and PABLO SHMERKINa2

a1 Microsoft Research, Redmond and Departments of Statistics and Mathematics, University of California, Berkeley, USA (email: peres@stat.berkeley.edu)

a2 Departments of Mathematics and Statistics, University of Jyväskylä, Finland (email: shmerkin@maths.jyu.fi)

Abstract

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)