Resonance between Cantor sets

Abstract

Let Ca 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, thenwhere dim is 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.

A family of smooth Cantor sets

“…Nazarov et al [NPS09] determined that the correlation dimension of µ r * µ s is min d r + d s , 1 whenever log r/ log s is irrational. Shmerkin informed us that in a joint work with Hochman [HS09] they generalize the work on sums of Cantor sets [PS09] and their methods imply that the dimension of the convolution H p * H p is min dim C p + dim C p , 1 whenever p/p is irrational. This in turns implies that for these parameters formula (1) holds.…”

“…It was improved in [PS00] as we mentioned above. Recently Peres and Shmerkin [PS09] found the exceptional set for K = A s : when dim A r + dim A s ≤ 1, equality holds if and only if log r/ log s is irrational. This condition also appears in the study of the topological structure of the sumset when dim A s + dim A r > 1; see Mendes and Oliveira [MO94] and Cabrelli et al [CHM02].…”

Absolute continuity of complex Bernoulli convolutions

“…There exists a self-similar set A 1 Ă A which is generated by a homogeneous IFS, satisfies the open set condition, and has dimension ą 1: this follows e.g. from [17,Lemma 3.6] and [20,Proposition 6]. Hence the claim follows by applying Theorem C to A 1 .…”

Fractal Geometry and Stochastics V

“…Peres and Shmerkin [74] proved (8.3) in the plane without requiring any separation condition on the IFS. To show this they set up a discrete version of Marstrand's projection theorem to construct a tree of intervals in the subspace (line) V followed by an application of Weyl's equidistribution theorem.…”