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Esa Järvenpää

Publications and source records attributed to Esa Järvenpää.

At least 19 recordsLinked to original sources
arXiv2025

Hitting Probabilities and the Ekstr{ö}m-Persson conjecture

We consider the Ekst\''om-Persson conjecture concerning the value of the Hausdorff dimension of random covering sets formed by balls with radii $(k^{-α})_{k=1}^\infty$ and centres chosen independently at random according to an arbitrary Borel probability measure $μ$ on $\mathbb{R}^d$. The conjecture has been solved positively in the case $\frac 1α\le \overline{\dim}_H μ$, where $\overline{\dim}_H μ$ stands for the upper Hausdorff dimension of $μ$. In this paper, we develop a new approach in order to answer the full conjecture, proving in particular that the conjectured value is only a lower bound for the dimension. Our approach opens the way to study more general limsup sets, and has consequences on the so-called hitting probability questions. For instance, we are able to determine whether and what part of a deterministic analytic set can be hit by random covering sets formed by open sets.

math.PR↗
arXiv2024

Ekström-Persson conjecture regarding random covering sets

We consider the Hausdorff dimension of random covering sets generated by balls and general measures in Euclidean spaces. We prove, for a certain parameter range, a conjecture by Ekström and Persson concerning the exact value of the dimension in the special case of radii $(n^{-α})_{n=1}^\infty$. For generating balls with an arbitrary sequence of radii, we find sharp bounds for the dimension and show that the natural extension of the Ekström-Persson conjecture is not true in this case. Finally, we construct examples demonstrating that there does not exist a dimension formula involving only the lower and upper local dimensions of the measure and a critical parameter determined by the sequence of radii.

math.CA↗
arXiv2022

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

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 open set condition. The bound is valid for all visible parts and it depends on 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.

math.DS↗
arXiv2021

Fractal percolation is unrectifiable

We show that there exists $0<α_0<1$ (depending on the parameters) such that the fractal percolation is almost surely purely $α$-unrectifiable for all $α>α_0$.

math.PR↗
arXiv2017

Asymptotically sharp dimension estimates for $k$-porous sets

In $\mathbb{R}^n$, we establish an asymptotically sharp upper bound for the upper Minkowski dimension of $k$-porous sets having holes of certain size near every point in $k$ orthogonal directions at all small scales. This bound tends to $n-k$ as $k$-porosity tends to its maximum value.

math.CA↗
arXiv2017

Packing dimension and Ahlfors regularity of porous sets in metric spaces

Let $X$ be a metric measure space with an $s$-regular measure $μ$. We prove that if $A\subset X$ is $\varrho$-porous, then $\dim_{\mathrm{p}}(A)\le s-c\varrho^s$ where $\dim_{\mathrm{p}}$ is the packing dimension and $c$ is a positive constant which depends on $s$ and the structure constants of $μ$. This is an analogue of a well known asymptotically sharp result in Euclidean spaces. We illustrate by an example that the corresponding result is not valid if $μ$ is a doubling measure. However, in the doubling case we find a fixed $N\subset X$ with $μ(N)=0$ such that $\dim_{\mathrm{p}}(A)\le\dim_{\mathrm{p}}(X)-c(\log\tfrac 1\varrho)^{-1}\varrho^t$ for all $\varrho$-porous sets $A\subset X\setminus N$. Here $c$ and $t$ are constants which depend on the structure constant of $μ$. Finally, we characterize uniformly porous sets in complete $s$-regular metric spaces in terms of regular sets by verifying that $A$ is uniformly porous if and only if there is $t<s$ and a $t$-regular set $F$ such that $A\subset F$.

math.CA↗
arXiv2016

Porosities of Mandelbrot Percolation

We study porosities in the Mandelbrot percolation process. We show that, almost surely at almost all points with respect to the natural measure, the 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 lower porosities of the set and the natural measure are equal to zero, whereas the upper porosities obtain their maximum values.

math.PR↗
arXiv2016

Hitting probabilities of random covering sets in tori and metric spaces

We provide sharp lower and upper bounds for the Hausdorff dimension of the intersection of a typical random covering set with a fixed analytic set both in Ahlfors regular metric spaces and in the $d$-dimensional torus. In metric spaces, we consider covering sets generated by balls and, in the torus, we deal with general analytic generating sets.

math.PR↗
arXiv2015

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

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.

math.DS↗
arXiv2015

Dimensions of random covering sets in Riemann manifolds

Let ${\pmb M}$, ${\pmb N}$ and ${\pmb K}$ be $d$-dimensional Riemann manifolds. Assume that ${\bf A}:=(A_n)_{n\in{\Bbb N}}$ is a sequence of Lebesgue measurable subsets of ${\pmb M}$ satisfying a necessary density condition and ${\bf x}:=(x_n)_{n\in {\Bbb N}}$ is a sequence of independent random variables which are distributed on ${\pmb K}$ 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 ${\bf E}({\bf x},{\bf A}):=\limsup_{n\to\infty}A_n(x_n)\subset {\pmb N}$. Here $A_n(x_n)$ is a diffeomorphic image of $A_n$ depending on $x_n$. We also verify that the packing dimensions of ${\bf E}({\bf x},{\bf A})$ equal $d$ almost surely.

math.CA↗
arXiv2014

Random affine code tree fractals and Falconer-Sloan condition

We calculate the almost sure dimension for a general class of random affine code tree fractals in $\mathbb R^d$. The result is based on a probabilistic version of the Falconer-Sloan condition $C(s)$ introduced in \cite{FS}. We verify that, in general, systems having a small number of maps do not satisfy condition $C(s)$. However, there exists a natural number $n$ such that for typical systems the family of all iterates up to level $n$ satisfies condition $C(s)$.

math.DS↗
arXiv2013

Hausdorff dimension of affine random covering sets in torus

We calculate the almost sure Hausdorff dimension of the random covering set $\limsup_{n\to\infty}(g_n + ξ_n)$ in $d$-dimensional torus $\mathbb T^d$, where the sets $g_n\subset\mathbb T^d$ are parallelepipeds, or more generally, linear images of a set with nonempty interior, and $ξ_n\in\mathbb T^d$ are independent and uniformly distributed random points. The dimension formula, derived from the singular values of the linear mappings, holds provided that the sequences of the singular values are decreasing.

math.PR↗
arXiv2012

Besicovitch-Federer projection theorem and geodesic flows on Riemann surfaces

We extend the Besicovitch-Federer projection theorem to transversal families of mappings. As an application we show that on a certain class of Riemann surfaces with constant negative curvature and with boundary, there exist natural 2-dimensional measures invariant under the geodesic flow having 2-dimensional supports such that their projections to the base manifold are 2-dimensional but the supports of the projections are Lebesgue negligible. In particular, the union of complete geodesics has Hausdorff dimension 2 and is Lebesgue negligible.

math.CA↗
arXiv2012

Hausdorff dimension and non-degenerate families of projections

We study parametrized families of orthogonal projections for which the dimension of the parameter space is strictly less than that of the Grassmann manifold. We answer the natural question of how much the Hausdorff dimension may decrease by verifying the best possible lower bound for the dimension of almost all projections of a finite measure. We also show that a similar result is valid for smooth families of maps from $n$-dimensional Euclidean space to $m$-dimensional one.

math.CA↗
arXiv2012

Dimensions of random affine code tree fractals

We calculate the almost sure Hausdorff dimension for a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous Markov constructions.

math.DS↗