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Marianna Csörnyei

Publications and source records attributed to Marianna Csörnyei.

11 recordsLinked to original sources
arXiv2026

Algorithmic Information Bounds for Distances and Orthogonal Projections

We introduce a new technique for proving bounds on the Kolmogorov complexity of geometric objects in Euclidean space, such as points and lines. We apply this technique to prove two theorems on algorithmic information theory, both of which have consequences for well-known problems in geometric measure theory. First, we show that for any point $x$ in the plane and any other point $y$ sufficiently independent of $x$, the distance between $x$ and $y$ retains at least half the complexity of the original point $x$. By the point-to-set principle of J. Lutz and N. Lutz, this yields an improved lower bound on the Hausdorff dimension of pinned distance sets, a topic closely related to Falconer's distance set conjecture. Second, we prove an analogous result for orthogonal projections: for any point $x$ in the plane and any line through the origin which is sufficiently independent of $x$, the projection of $x$ onto that line retains at least half the complexity of $x$. As a consequence, we obtain a generalization of a theorem of Bourgain on exceptional sets for orthogonal projections.

cs.CC↗
arXiv2025

On the modulus of continuity of functions whose image has positive measure, and metric embeddings into $\mathbb{R}^d$ without shrinking

A generalization of the classical Sard theorem in the plane is the following. Let $f$ be a function defined on a subset $A\subset{\mathbb R}^2$. If $f$ has modulus of continuity $ω(r)\lesssim r^2$, then $f(A)\subset{\mathbb R}$ has Lebesgue measure zero. Choquet claimed in \cite{Choquet} that this was a full characterization, i.e. for every $ω$ for which $ω(r)/r^2$ converges to $\infty$ as $r\to 0$, there is a counterexample. We disprove this by showing that the correct characterization, in $\mathbb{R}^d$, is $\int_{0}^{1} ω(r)^{-1/d}=\infty$. For the precise statement see Theorem 2. We obtain this as a special case of a more general result. We study which spaces $(X,ρ)$ can be embedded into ${\mathbb R}^d$ without decreasing any of the distances in $X$. That is, we ask the question whether there is an $f: X\to {\mathbb R}^d$ such that $\|f(x)-f(y)\|\ge ρ(x,y)$ for every $x,y\in X$. We study this problem for some very general distance functions $ρ$ (we do not even assume that it is a metric space, in particular, we do not assume that $ρ$ satisfies the triangle inequality), and find quantitative necessary and sufficient conditions under which such a mapping exists. We will obtain the characterization mentioned above as a special case of our metric embedding results, by choosing $X$ to be an interval in $\mathbb{R}$, and defining $ρ$ by putting $ρ(x,y)=r$ if $\|x-y\|=ω(r)$.

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↗
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

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↗
arXiv2014

Tangents of $σ$-finite Curves and Scaled Oscillation

We show that every continuous simple curve with $σ$-finite length has a tangent at positively many points. We also apply this result to functions with finite lower scaled oscillation; and study the validity of the results in higher dimension.

math.CA↗
arXiv2012

On Cantor sets and doubling measures

For a large class of Cantor sets on the real-line, we find sufficient and necessary conditions implying that a set has positive (resp. null) measure for all doubling measures of the real-line. We also discuss same type of questions for atomic doubling measures defined on certain midpoint Cantor sets.

math.CA↗
arXiv2007

Tube-measurability

Define an outer measure on R^n by taking the infimum, over all covers of the set by tubes, of the sum of the cross-sectional areas of the tubes. We show that the only measurable sets for this outer measure are its null sets and their complements.

math.CA↗