arXiv subjects

Kornélia Héra

Publications and source records attributed to Kornélia Héra.

7 recordsLinked to original sources

arXiv2025

Signed null sequences and Hausdorff dimension

We investigate the convergence of signed null sequences of the form \[ \sum_{n=1}^\infty \varepsilon_n a_n, \quad \varepsilon_n \in \{-1,1\}, \] where $(a_n)$ tends to zero in $\mathbb{R}^d$. Our main result shows that for any such sequence, the set of sign sequences yielding convergence has full Hausdorff dimension in the natural ultrametric topology. This answers a question of Mattila in the one-dimensional case, for which we provide an elementary proof. Moreover, if $(a_n)\notin \ell^1$ in one dimension, then for every $L\in\mathbb{R}$ the set of sign sequences with sum $L$ also has Hausdorff dimension $1$. In higher dimensions the analogous statement does not hold in full generality, but it is guaranteed if the sequence has $d$ linearly independent Lévy vectors.

math.CA↗

arXiv2023

Hausdorff dimension of Besicovitch sets of Cantor graphs

We consider the Hausdorff dimension of planar Besicovitch sets for rectifiable sets $Γ$, i.e. sets that contain a rotated copy of $Γ$ in each direction. We show that for a large class of Cantor sets $C$ and Cantor-graphs $Γ$ built on $C$, the Hausdorff dimension of any $Γ$-Besicovitch set must be at least $\min\left(2-s^2,\frac{1}{s}\right)$, where $s=\dim C$.

math.MG↗

arXiv2020

An improved bound for the dimension of $(α,2α)$-Furstenberg sets

We show that given $α\in (0, 1)$ there is a constant $c=c(α) > 0$ such that any planar $(α, 2α)$-Furstenberg set has Hausdorff dimension at least $2α+ c$. This improves several previous bounds, in particular extending a result of Katz-Tao and Bourgain. We follow the Katz-Tao approach with suitable changes, along the way clarifying, simplifying and/or quantifying many of the steps.

math.CA↗

arXiv2019

Hausdorff dimension of Furstenberg-type sets associated to families of affine subspaces

We show that if $B \subset \mathbb{R}^n$ and $E \subset A(n,k)$ is a nonempty collection of $k$-dimensional affine subspaces of $\mathbb{R}^n$ such that every $P \in E$ intersects $B$ in a set of Hausdorff dimension at least $α$ with $k-1 < α\leq k$, then $\dim B \geq α+\dim E/(k+1)$, where $\dim$ denotes the Hausdorff dimension. This estimate generalizes the well known Furstenberg-type estimate that every $α$-Furstenberg set in the plane has Hausdorff dimension at least $α+ 1/2$. More generally, we prove that if $B$ and $E$ are as above with $0 < α\leq k$, then $\dim B \geq α+(\dim E-(k-\lceil α\rceil)(n-k))/(\lceil α\rceil+1)$. We also show that this bound is sharp for some parameters. As a consequence, we prove that for any $1 \leq k<n$, the union of any nonempty $s$-Hausdorff dimensional family of $k$-dimensional affine subspaces of $\mathbb{R}^n$ has Hausdorff dimension at least $k+\frac{s}{k+1}$.

math.MG↗

arXiv2018

Small unions of affine subspaces and skeletons via Baire category

Our aim is to find the minimal Hausdorff dimension of the union of scaled and/or rotated copies of the $k$-skeleton of a fixed polytope centered at the points of a given set. For many of these problems, we show that a typical arrangement in the sense of Baire category gives minimal Hausdorff dimension. In particular, this proves a conjecture of R. Thornton. Our results also show that Nikodym sets are typical among all sets which contain, for every point x of R^n, a punctured hyperplane H\{x} through x. With similar methods we also construct a Borel subset of R^n of Lebesgue measure zero containing a hyperplane at every positive distance from every point.

math.MG↗

arXiv2018

The Kakeya problem for circular arcs

We prove that if a circular arc has angle short enough, then it can be continuously moved to any prescribed position within a set of arbitrarily small area.

math.MG↗

arXiv2018

Closed sets with the Kakeya property

We say that a planar set $A$ has the Kakeya property if there exist two different positions of $A$ such that $A$ can be continuously moved from the first position to the second within a set of arbitrarily small area. We prove that if $A$ is closed and has the Kakeya property, then the union of the nontrivial connected components of $A$ can be covered by a null set which is either the union of parallel lines or the union of concentric circles. In particular, if $A$ is closed, connected and has the Kakeya property, then $A$ can be covered by a line or a circle.

math.MG↗