arXiv subjects

Marianna Csornyei

Publications and source records attributed to Marianna Csornyei.

5 recordsLinked to original sources

arXiv2026

Hyperplane Incidences and Distance Sets in Higher Dimensions

We generalize Ren and Wang's incidence bound between points and lines in $\R^2$ \cite{RenWan23} to higher dimensions. We show how to use this incidence bound to improve the best known bound for Falconer's distance set problem in $\R^3$ and in $\R^4$. We show that if $d=3$ or $d=4$, and $E\subset \R^d$ is a Borel set of dimension $\dim_H(E) > d/2$, then \begin{equation*} \sup_{x\in E} \dim_H(Δ_x(E)) \geq 2/3, \end{equation*} where $Δ_x(E)$ is the pinned distance set of $E$ with respect to $x$. We also show how the incidence bound can be used to generalize the planar Furstenberg set bound, to sets in $\R^d$ that contain a $t$-dimensional set of hyperplanes, each of which contains an $s$-dimensional set of points, for any $d\ge 2$, $s \in (d-2, d-1]$ and $t \in (0, d]$.

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arXiv2026

Improved bounds for radial projections in the plane

We improve the best known lower bound for the dimension of radial projections of sets in the plane. We show that if $X,Y$ are Borel sets in $\R^2$, $X$ is not contained in any line and $\dim_H(X)>0$, then $$\sup\limits_{x\in X} \dim_H(π_x Y) \geq \min\left\{(\dim_H(Y) + \dim_H(X))/2, \dim_H(Y), 1\right\},$$ where $π_x Y$ is the radial projection of the set $Y$ from the point $x$.

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arXiv2025

Bounding the dimension of exceptional sets for orthogonal projections

It is well known that if $A \subseteq \mathbb{R}^n$ is an analytic set of Hausdorff dimension $a$, then $\dim_H(π_VA)=\min\{a,k\}$ for a.e.\ $V\in G(n,k)$, where $G(n,k)$ denotes the set of all $k$-dimensional subspaces of $\mathbb{R}^n$ and $π_V$ is the orthogonal projection of $A$ onto $V$. In this paper we study how large the exceptional set \begin{equation*} \{V\in G(n,k) \mid \dim_H(π_V A) < s\} \end{equation*} can be for a given $s\le\min\{a,k\}.$ We improve previously known estimates on the dimension of the exceptional set, and we show that our estimates are sharp for $k=1$ and for $k=n-1$. Hence we completely resolve this question for $n=3$.

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arXiv2021

Scaled Oscillation and Level Sets

We study the size and regularity properties of level sets of continuous functions with bounded upper-scaled and lower-scaled oscillation.

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arXiv2006

Universal singular sets in the calculus of variations

For regular one-dimensional variational problems, Ball and Nadirashvilli introduced the notion of the universal singular set of a Lagrangian L and established its topological negligibility. This set is defined to be the set of all points in the plane through which the graph of some absolutely continuous L-minimizer passes with infinite derivative. Motivated by Tonelli's partial regularity results, the question of the size of the universal singular set in measure naturally arises. Here we show that universal singular sets are characterized by being essentially purely unrectifiable -- that is, they intersect most Lipschitz curves in sets of zero length and that any compact purely unrectifiable set is contained within the universal singular set of some smooth Lagrangian with given superlinear growth. This gives examples of universal singular sets of Hausdorff dimension two, filling the gap between previously known one-dimensional examples and Sychev's result that universal singular sets are Lebesgue null. We show that some smoothness of the Lagrangian is necessary for the topological size estimate, and investigate the relationship between growth of the Lagrangian and the existence of (pathological) rectifiable pieces in the universal singular set. We also show that Tonelli's partial regularity result is stable in that the energy of a `near' minimizer u over the set where it has large derivative is controlled by the how far u is from being a minimizer.

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