A Hausdorff measure version of W.M. Schmidt’s inhomogeneous, linear forms theorem in metric number theory is established. The key ingredient is a ‘slicing’ technique motivated by a standard result in geometric measure theory. In short, ‘slicing’ together with the Mass Transference Principle [3] allows us to transfer Lebesgue measure theoretic statements for lim sup sets associated with linear f...

In [Berarducci-Otero, 2004], the authors define an analogue of Lebesgue measure for bounded definable subsets of an o-minimal structure K , which expands a real closed field. Our aim is defining the area of a bounded definable d-dimensional subsets of Kn. Different possible definitions for such area are possible: we will prove that some of them are equivalent. On the reals, there are many diffe...

For $S_g(x,y)=x-g(y), x,y\in\mathbb{R}^n, g\in O(n),$ we investigate the Lebesgue measure and Hausdorff dimension of $S_g(A)$ given $A$, both for general Borel subsets $\mathbb{R}^{2n}$ product sets.

We show that for every Lebesgue measurable function f : [0, 1] → R there exists a compact set C of Hausdorff dimension 1/2 such that f is of bounded variation on C, and there exist compact sets Cα of Hausdorff dimension 1−α such that f is Hölder-α on Cα (0 < α < 1). These answer questions of M. Elekes, which were open even for continuous functions f . Our proof goes by defining a discrete Hausd...

We discuss the weak form of the Ramberg/Osgood equations for nonlinear elastic materials on a 3-dimensional domain and show that the stress tensor is Hölder continuous on an open subset whose complement is of Lebesgue-measure zero. We also give an estimate for the Hausdorff-dimension of the singular set.

Using Voiculescu’s notion of a matricial microstate we introduce fractal dimensions and entropy for finite sets of self-adjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of free Hausdorff dimension is an algebrai...

Let Γ be a closed set in Rn with the Lebesgue measure |Γ| = 0. The first aim of the paper is to give a Fourier analytical characterization of the Hausdorff dimension of Γ. Let 0 < d < n. If there exist a Borel measure μ with supp μ ⊂ Γ and constants c1 > 0 and c2 > 0 such that c1r ≤ μ(B(x, r)) ≤ c2r for all 0 < r < 1 and all x ∈ Γ, where B(x, r) is a ball with centre x and radius r, then Γ is c...