arXiv subjects

M. Laczkovich

Publications and source records attributed to M. Laczkovich.

5 recordsLinked to original sources

arXiv2026

Translating measurable sets

We prove that if $A,B$ are compact subsets of $\mathbb{R}$ such that the upper density of $B$ is positive at every point of $B$, then there is a closed null set $N\subset A$ such that $N+B=A+B$. As a corollary we find that if $A,B\subset \mathbb{R}$ are measurable, and every null subset $N$ of $A$ can be translated into $B$ (that is, if $B$ contains a suitable translate of $N$), then there is a null set $N_0$ such that $A\setminus N_0$ can be translated into $B$. The topic is related to some consistency results of the theory of additive properties of the reals.

math.CA↗

arXiv2026

Graph-null sets

We say that a plane set $A$ is {\it graph-null,} if there is a function $g\colon [0,1] \to \mathbb{R}$ such that $λ_2 (A+{\rm graph}\, g)=0$. A plane set $A$ has the {\it translational Kakeya property} if, for every translated copy $A'$ of $A$ and for every $ε>0$, there is a finite sequence of vertical and horizontal translations bringing $A$ to $A'$ such that the area touched during the horizontal translations is less than $ε$. These properties are equivalent if $A$ is compact. We show that the graph of every absolutely continuous function is graph-null. Also, the graph of a typical continuous function is graph-null. Therefore, there are nowhere differentiable continuous functions whose graphs are graph-null. Still, we show that there exists a continuous function whose graph is not graph-null.

math.CO↗

arXiv2021

A superposition theorem of Kolmogorov type for bounded continuous functions

Let $C({\mathbb R}^n)$ denote the set of real valued continuous functions defined on ${\mathbb R}^n$. We prove that for every $n\ge 2$ there are positive numbers $λ_1 , \ldots , λ_n$ and continuous functions $ϕ_1 ,\ldots , ϕ_m \in C({\mathbb R})$ with the following property: for every bounded and continuous $f\in C( {\mathbb R}^n )$ there is a continuous function $g\in C({\mathbb R} )$ such that $$f(x)=\sum_{q=1}^m g\left( \sum_{p=1}^n λ_p ϕ_q (x_p ) \right)$$ for every $x=(x_1 ,\ldots , x_n )\in {\mathbb R}^n$. Consequently, every $f\in C({\mathbb R}^n)$ can be obtained from continuous functions of one variable using compositions and additions.

math.CA↗

arXiv2020

Irregular tilings of regular polygons with similar triangles

We say that a triangle $T$ tiles a polygon $A$, if $A$ can be dissected into finitely many nonoverlapping triangles similar to $T$. We show that if $N>42$, then there are at most three nonsimilar triangles $T$ such that the angles of $T$ are rational multiples of $π$ and $T$ tiles the regular $N$-gon. A tiling into similar triangles is called regular, if the pieces have two angles, $\al$ and $\be$, such that at each vertex of the tiling the number of angles $\al$ is the same as that of $\be$. Otherwise the tiling is irregular. It is known that for every regular polygon $A$ there are infinitely many triangles that tile $A$ regularly. We show that if $N>10$, then a triangle $T$ tiles the regular $N$-gon irregularly only if the angles of $T$ are rational multiples of $π$. Therefore, the numbers of triangles tiling the regular $N$-gon irregularly is at most three for every $N>42$.

math.MG↗

arXiv1994

Measurability of functions with approximately continuous vertical sections and measurable horizontal sections

A function f:R -> R is approximately continuous iff it is continuous in the density topology, i.e., for any ordinary open set U the set E=f^{-1}(U) is measurable and has Lebesgue density one at each of its points. Denjoy proved that approximately continuous functions are Baire 1., i.e., pointwise For any f:R^2 -> R define f_x(y) = f^y(x) = f(x,y). A function f:R^2 -> R is separately continuous if f_x and f^y are continuous for every x,y in R. Lebesgue in his first paper proved that any separately continuous function is Baire 1. Sierpinski showed that there exists a nonmeasurable f:R^2 -> R which is separately Baire 1. In this paper we prove: Thm 1. Let f:R^2 -> R be such that f_x is approximately continuous and f^y is Baire 1 for every x,y in R. Then f is Baire 2. Thm 2. Suppose there exists a real-valued measurable cardinal. Then for any function f:R^2 -> R and countable ordinal i, if f_x is approximately continuous and f^y is Baire i for every x,y in R, then f is Baire i+1 as a function of two variables. Thm 3. (i) Suppose that R can be covered by omega_1 closed null sets. Then there exists a nonmeasurable function f:R^2 -> R such that f_x is approximately continuous and f^y is Baire 2 for every x,y in R. (ii) Suppose that R can be covered by omega_1 null sets. Then there exists a nonmeasurable function f:R^2 -> R such that f_x is approximately continuous and f^y is Baire 3 for every x,y in R. Thm 4. In the random real model for any function f:R^2 -> R if f_x is approximately continuous and f^y is measurable for every x,y in R, then f is measurable as a function of two variables.

math.LO↗