Resonance between Cantor sets

By Yuval Peres, Pablo Shmerkin. Ergodic Theory and Dynamical Systems, 2007

127 citations recorded by the source corpus.

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 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.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ed4592f68443002f3cd8ed25648d24bb19900907

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Citations in the corpus

A proof of Furstenberg's conjecture on the intersections of ⨯p- and ⨯q-invariant sets (2016, background): “Recently, there has been considerable interest in the study of projections of dynamically defined Cantor sets, see for instance the survey paper of Shmerkin [34] and the references therein for more details.”
On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions (2013, background): “Peres and Shmerkin [21] proved that, when K is a scaled copy of Aλ′ , there are only countably many exceptions for the dimension result (uniformly in the scaling): dimH(Aλ + rAλ′) = min(dimH(Aλ) + dimH(Aλ′), 1) if log λ log λ′ /∈ Q, r 6= 0.”
Sixty years of fractal projections (2014, methodology): “That the open set condition is not essential follows since, for all > 0, we can use a Vitali argument to set up a new IFS, consisting of compositions of the fi, that satisfies SSC, with attractor E ′ ⊂ E such that dimHE ′ > dimHE− ; we can also ensure that the new IFS has dense rotations if the orig”
On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions (2016, background): “It was observed in several previous works, going back at least to [38], that methods devised to study geometric properties of cartesian products of linear self-similar sets and measures often can also be applied to the study of selfsimilar sets and measures on the plane.”
Absolute continuity of self-similar measures, their projections and convolutions (2014, background): “se Gδ). Nevertheless, recently there has been much interest in improving general projection results for speciﬁc class of sets and measures; in particular, for self-similar sets and measures: see e.g. [22, 18, 13, 11, 5, 7]. However, all of these papers deal with the dimension part, and the technique”
Dynamics on fractals and fractal distributions (2010, result): “In particular, this work was motivated by some recent results by Furstenberg [14], Peres and Shmerkin [27], and Hochman and Shmerkin [19], on the geometry of measures which arise from certain combinatorial constructions or as invariant measures for certain dynamics.”
Local entropy averages and projections of fractal measures (2009, methodology): “ese hypotheses he shows that if X,Y are the attractors then the sumset X + Y, which is the projection of X × Y under π(x,y) = x + y, has the expected dimension. More recently Y. Peres and P. Shmerkin [31] solved the problem for projections ofX× Y when X,Y are attractors of linear IFSs satisfying an ”
On the structures of generating iterated function systems of Cantor sets (2009, unknown): “Shmerkin [22] told the authors that an argument using the results on sum of Cantor sets in [20] can prove the Logarithmic Commensurability Theorem for any IFS in R of the form {ρkix+ ci}i=1, ki ∈ N, under the assumption that the Hausdorff dimension of the attractor is less than 1 and coincides with ”
Equidistribution from fractal measures (2013, unknown): “Following terminology of Peres and Shmerkin [42], we say that measures µ, ν ∈ P(R) resonate if dimµ ∗ ν min{1,dimµ+ dim ν} (11) otherwise they dissonate.”