We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We apply this to prove the following conjecture of Furstenberg: if X, Y ⊆ [0, 1] are closed and invariant, respectively, under ×m mod 1 and ×n mod 1, where m, n are not powers of the same integer, then, for any t = 0,A similar result holds for invariant measures and gives a simple proof of the Rudolph-Johnson theorem. Our methods also apply to many other classes of conformal fractals and measures. As another application, we extend and unify results of Peres, Shmerkin and Nazarov, and of Moreira, concerning projections of products of self-similar measures and Gibbs measures on regular Cantor sets. We show that under natural irreducibility assumptions on the maps in the IFS, the image measure has the maximal possible dimension under any linear projection other than the coordinate projections. We also present applications to Bernoulli convolutions and to the images of fractal measures under differentiable maps.
We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the L q dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's longstanding conjecture on the dimension of the intersections of ×p and ×q-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an L q density for all finite q, outside of a zero-dimensional set of exceptions.The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to L q norms, and likewise relies on an inverse theorem for the decay of L q norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemerédi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.
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.
We give a fractal-geometric condition for a measure on [0, 1] to be supported on points x that are normal in base n, i.e. such that {n k x} k∈N equidistributes modulo 1. This condition is robust under C 1 coordinate changes, and it applies also when n is a Pisot number rather than an integer. As applications we obtain new results (and strengthen old ones) about the prevalence of normal numbers in fractal sets, and new results on measure rigidity, specifically completing Host's theorem to multiplicatively independent integers and proving a Rudolph-Johnson-type theorem for certain pairs of beta transformations.
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