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Resonance between Cantor sets

Published online by Cambridge University Press:  01 February 2009

YUVAL PERES  and
PABLO SHMERKIN
YUVAL PERES
Affiliation:
Microsoft Research, Redmond and Departments of Statistics and Mathematics, University of California, Berkeley, USA (email: peres@stat.berkeley.edu)
PABLO SHMERKIN
Affiliation:
Departments of Mathematics and Statistics, University of Jyväskylä, Finland (email: shmerkin@maths.jyu.fi)
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Abstract

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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 dimis 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.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 1 , February 2009 , pp. 201 - 221
Copyright
Copyright © 2008 Cambridge University Press

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