Porosities and dimensions of measures

We introduce a concept of porosity for measures and study relations between dimensions and porosities for two classes of measures: measures on $R^n$ which satisfy the doubling condition and strongly porous measures on $R$.Comment: Jarvenpaa = J\"arvenp\"a\"

Hausdorff dimension of limsup sets of rectangles in the Heisenberg group

Abstract The almost sure value of the Hausdorff dimension of limsup sets generated by randomly distributed rectangles in the Heisenberg group is computed in terms of directed singular value functions

Porosities of Mandelbrot percolation

Abstract We study porosities in the Mandelbrot percolation process using a notion of porosity that is based on the construction geometry. We show that, almost surely at almost all points with respect to the natural measure, the construction-based mean porosities of the set and the natural measure exist and are equal to each other for all parameter values outside of a countable exceptional set. As a corollary, we obtain that, almost surely at almost all points, the regular lower porosities of the set and the natural measure are equal to zero, whereas the regular upper porosities reach their maximum values

On dimensions of visible parts of self-similar sets with finite rotation groups

Abstract We derive an upper bound for the Assouad dimension of visible parts of self-similar sets generated by iterated function systems with finite rotation groups and satisfying the weak separation condition. The bound is valid for all visible parts and it depends on the direction and the penetrable part of the set, which is a concept defined in this paper. As a corollary, we obtain in the planar case that if the projection is a finite or countable union of intervals then the visible part is 1-dimensional. We also prove that the Assouad dimension of a visible part is strictly smaller than the Hausdorff dimension of the set provided the projection contains interior points. Our proof relies on Furstenberg’s dimension conservation principle for self-similar sets

Random affine code tree fractals:Hausdorff and affinity dimensions and pressure

Abstract We prove that for random affine code tree fractals the affinity dimension is almost surely equal to the unique zero of the pressure function. As a consequence, we show that the Hausdorff, packing and box counting dimensions of such systems are equal to the zero of the pressure. In particular, we do not presume the validity of the Falconer-Sloan condition or any other additional assumptions which have been essential in all the previously known results

Hausdorff dimension of limsup sets of random rectangles in products of regular spaces

Abstract The almost sure Hausdorff dimension of the limsup set of randomly distributed rectangles in a product of Ahlfors regular metric spaces is computed in terms of the singular value function of the rectangles

Dimensions of random covering sets in Riemann manifolds

Abstract Let 𝐌, 𝐍 and 𝐊 be d-dimensional Riemann manifolds. Assume that 𝐀 := (An)n∈ℕ is a sequence of Lebesgue measurable subsets of 𝐌 satisfying a necessary density condition and 𝐱 := (xn)n∈ℕ is a sequence of independent random variables, which are distributed on 𝐊 according to a measure, which is not purely singular with respect to the Riemann volume. We give a formula for the almost sure value of the Hausdorff dimension of random covering sets 𝐄(𝐱, 𝐀) := lim supn→∞An(xn) ⊂ 𝐍. Here, An(xn) is a diffeomorphic image of An depending on xn. We also verify that the packing dimensions of 𝐄(𝐱, 𝐀) equal d almost surely

Fractal percolation is unrectifiable

Abstract We show that there exists 0 < α₀ < 1 (depending on the parameters) such that the fractal percolation is almost surely purely α-unrectifiable for all α > α₀

Visible parts and dimensions

We study the visible parts of subsets of n-dimensional Euclidean space: a point a of a compact set A is visible from an affine subspace K of Rn, if the line segment joining PK(a) to a only intersects A at a (here PK denotes projection onto K). The set of all such points visible from a given subspace K is called the visible part of A from K. We prove that if the Hausdorff dimension of a compact set is at most n−1, then the Hausdorff dimension of a visible part is almost surely equal to the Hausdorff dimension of the set. On the other hand, provided that the set has Hausdorff dimension larger than n − 1, we have the almost sure lower bound n − 1 for the Hausdorff dimensions of visible parts. We also investigate some examples of planar sets with Hausdorff dimension bigger than 1. In particular,we prove that for quasi-circles in the plane all visible parts have Hausdorff dimension equal to 1