Resonance between Cantor sets

TL;DR

This paper investigates the Hausdorff dimension of sums of Cantor sets and self-similar sets, revealing conditions under which the dimension behaves as expected or exhibits resonance phenomena, with implications for projections of fractal sets.

Contribution

It establishes a criterion linking irrationality of log ratios to dimension sums and introduces the concept of algebraic resonance for self-similar sets with smaller-than-expected sum dimensions.

Findings

Dimension of sum of Cantor sets equals the sum or 1 when log ratios are irrational.

Resonance occurs when the sum dimension is smaller, implying algebraic relations among contraction ratios.

New results on projections of planar self-similar sets with irrational rotations.

Abstract

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.