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Let K ⊂ R d be a self-similar or self-affine set, let µ be a self-similar or self-affine measure on it, and let G be the group of affine maps, similitudes, isometries or translations of R d. Under various assumptions (such as separation conditions or we assume that the transformations are small perturbations or that K is a so called Sierpi´nski sponge) we(More)
In an earlier paper we introduced a new concept of dimension for metric spaces, the so called topological Hausdor dimension. For a compact metric space K let dimH K and dimtH K denote its Hausdor and topological Hausdor dimension, respectively. We proved that this new dimension describes the Hausdor dimension of the level sets of the generic continuous(More)
Let G be an abelian Polish group, e.g. a separable Banach space. A subset X ⊂ G is called Haar null (in the sense of Christensen) if there exists a Borel set B ⊃ X and a Borel probability measure µ on G such that µ(B + g) = 0 for every g ∈ G. The term shy is also commonly used for Haar null, and co-Haar null sets are often called prevalent. Answering an old(More)
A subset X of a Polish group G is called Haar null if there exists a Borel set B ⊃ X and Borel probability measure µ on G such that µ(gBh) = 0 for every g, h ∈ G. We prove that there exist a set X ⊂ R that is not Lebesgue null and a Borel probability measure µ such that µ(X + t) = 0 for every t ∈ R. This answers a question from David Fremlin's problem list(More)
We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in ZF C and even in the theory ZF C + c = ω 2 if the number of pieces can be(More)
Cain, Clark and Rose dened a function f : R n → R to be vertically rigid if graph(cf) is isometric to graph(f) for every c = 0. It is horizontally rigid if graph(f (c x)) is isometric to graph(f) for every c = 0 (see [1]). In [2] the authors of the present paper settled Jankovi¢'s conjecture by showing that a continuous function of one variable is(More)
The main goal of this note is to prove the following theorem. If An is a sequence of measurable sets in a σ-nite measure space (X, A, µ) that covers µ-a.e. x ∈ X innitely many times, then there exists a sequence of integers ni of density zero so that An i still covers µ-a.e. x ∈ X innitely many times. The proof is a probabilistic construction. As an(More)
Let K ⊂ R d be a self-similar or self-affine set, let µ be a self-similar or self-affine measure on it, and let G be the group of affine maps, similitudes, isometries or translations of R d. Under various assumptions (such as separation conditions or we assume that the transformations are small perturbations or that K is a so called Sierpi´nski sponge) we(More)
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