arXiv subjects
Publications and source records attributed to Fredrik Ekström.
The Hausdorff dimension of the set of points that are covered infinitely many times by a sequence of randomly distributed balls in the unit cube can be expressed in terms of the sizes of the balls. This note presents a new proof of the formula.
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.
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.
We prove bounds for the almost sure value of the Hausdorff dimension of the limsup set of a sequence of balls in $\mathbf{R}^d$ whose centres are independent, identically distributed random variables. The formulas obtained involve the rate of decrease of the radii of the balls and multifractal properties of the measure according to which the balls are distributed, and generalise formulas that are known to hold for particular classes of measures.
Given a compact set of real numbers, a random $C^{m + α}$-diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number $s$, almost surely has Fourier dimension greater than or equal to $s / (m + α)$. This is used to show that every Borel subset of the real numbers of Hausdorff dimension $s$ is $C^{m + α}$-equivalent to a set of Fourier dimension greater than or equal to $s / (m + α)$. In particular every Borel set is diffeomorphic to a Salem set, and the Fourier dimension is not invariant under $C^m$-diffeomorphisms for any $m$.
We give a sufficient condition for the Fourier dimension of a countable union of sets to equal the supremum of the Fourier dimensions of the sets in the union, and show by example that the Fourier dimension is not countably stable in general. A natural approach to finite stability of the Fourier dimension for sets would be to try to prove that the Fourier dimension for measures is finitely stable, but we give an example showing that it is not in general. We also describe some situations where the Fourier dimension for measures is stable or is stable for all but one value of some parameter. Finally we propose a way of modifying the definition of the Fourier dimension so that it becomes countably stable, and show that a measure has modified Fourier dimension greater than or equal to $s$ if and only if it annihilates all sets with modified Fourier dimension less than $s$.
The Fourier dimension is not in general stable under finite unions of sets. Moreover, the stability of the Fourier dimension on particular pairs of sets is independent from the stability of the compact Fourier dimension.
Various notions of "zooming in" on measures exist in the literature and the scenery flow is one of them. It is of interest to describe the joint asymptotics of the scenery flows generated by a measure and the measure transported by a local diffeomorphism. We give both sufficient and necessary conditions for the scenery distributions to be asymptotic and provide some examples.