arXiv subjects

Gáspár Vértesy

Publications and source records attributed to Gáspár Vértesy.

9 recordsLinked to original sources

arXiv2024

Better estimates of Hölder thickness of fractals

Dimensions of level sets of generic continuous functions and generic Hölder functions defined on a fractal $F$ encode information about the geometry, ``the thickness" of $F$. While in the continuous case this quantity is related to a reasonably tame dimension notion which is called the topological Hausdorff dimension of $F$, the Hölder case seems to be highly nontrivial. A number of earlier papers attempted to deal with this problem, carrying out investigation in the case of Hausdorff dimension and box dimension. In this paper we continue our study of the Hausdorff dimension of almost every level set of generic $1$-Hölder-$ α$ functions, denoted by $D_{*}( α, F)$. We substantially improve previous lower and upper bounds on $D_{*}( α, Δ)$, where $Δ$ is the Sierpiński triangle, achieving asymptotically equal bounds as $α\to 0+$. Using a similar argument, we also give an even stronger lower bound on the generic lower box dimension of level sets. Finally, we construct a connected fractal $F$ on which there is a phase transition of $D_{*}( α, F)$, thus providing the first example exhibiting this behavior.

math.CA↗

arXiv2022

Generic Hölder level sets and fractal conductivity

Hausdorff dimensions of level sets of generic continuous functions defined on fractals can give information about the "thickness/narrow cross-sections'' of a "network" corresponding to a fractal set, $F$. This lead to the definition of the topological Hausdorff dimension of fractals. In this paper we continue our study of the level sets of generic $1$-Hölder-$α$ functions. While in a previous paper we gave the initial definitions and established some properties of these generic level sets, in this paper we provide numerical estimates in the case of the Sierpiński triangle. These calculations give better insight and illustrate why can one think of these generic $1$-Hölder-$α$ level sets as something measuring "thickness/narrow cross-sections/conductivity'' of a fractal "network". We also give an example for the phenomenon which we call phase transition for $D_{*}(α, F)$. This roughly means that for a certain lower range { of $α$s } only the geometry of $F$ determines $D_{*}(α, F)$ while for larger values the Hölder exponent, $α$ also matters.

math.CA↗

arXiv2022

Generic Hölder level sets on Fractals

Hausdorff dimensions of level sets of generic continuous functions defined on fractals were considered in two papers by R. Balka, Z. Buczolich and M. Elekes. In those papers the topological Hausdorff dimension of fractals was defined. In this paper we start to study level sets of generic $1$-Hölder-$α$ functions defined on fractals. This is related to some sort of "thickness", "conductivity" properties of fractals. The main concept of our paper is $D_{*}(α, F)$ which is the essential supremum of the Hausdorff dimensions of the level sets of a generic $1$-Hölder-$α$ function defined on the fractal $F$. We prove some basic properties of $D_{*}(α, F)$, we calculate its value for an example of a "thick fractal sponge", we show that for connected self similar sets $D_{*}(α, F)$ it equals the Hausdorff dimension of almost every level in the range of a generic $1$-Hölder-$α$ function.

math.CA↗

arXiv2021

Strong one-sided density without uniform density

In this paper we give an example of a closed, strongly one-sided dense set which is not of uniform density type. We also show that there is a set of uniform density type which is not of strong uniform density type.

math.CA↗

arXiv2021

Big and little Lipschitz one sets

Given a continuous function $f: {\mathbb R}\to {\mathbb R}$ we denote the so-called "big Lip" and "little lip" functions by $ {\mathrm {Lip}} f$ and $ {\mathrm {lip}} f$ respectively}. In this paper we are interested in the following question. Given a set $E {\subset} {\mathbb R}$ is it possible to find a continuous function $f$ such that $ {\mathrm {lip}} f=\mathbf{1}_E$ or $ {\mathrm {Lip}} f=\mathbf{1}_E$? For monotone continuous functions we provide the rather straightforward answer. For arbitrary continuous functions the answer is much more difficult to find. We introduce the concept of uniform density type (UDT) and show that if $E$ is $G_δ$ and UDT then there exists a continuous function $f$ satisfying $ {\mathrm {Lip}} f =\mathbf{1}_E$, that is, $E$ is a $ {\mathrm {Lip}} 1$ set. In the other direction we show that every ${\mathrm {Lip}} 1$ set is $G_δ$ and weakly dense. We also show that the converse of this statement is not true, namely that there exist weakly dense $G_{δ}$ sets which are not $ {\mathrm {Lip}} 1$. We say that a set $E\subset \mathbb{R}$ is ${\mathrm {lip}} 1$ if there is a continuous function $f$ such that ${\mathrm {lip}} f=\mathbf{1}_E$. We introduce the concept of strongly one-sided density and show that every ${\mathrm {lip}} 1$ set is a strongly one-sided dense $F_σ$ set.

math.CA↗

arXiv2020

Characterization of lip sets

We denote the local ``little" Lipschitz constant of a function $f: {\mathbb R}\to { {\mathbb R}}$ by $ {\mathrm{lip}}f$. In this paper we settle the following question: For which sets $E {\subset} { {\mathbb R}}$ is it possible to find a continuous function $f$ such that $ {\mathrm{lip}}f=\mathbf{1} _E$? In an earlier paper we introduced the concept of strongly one-sided dense sets. Our main result characterizes $ {\mathrm{lip}}1$ sets as countable unions of closed sets which are strongly one-sided dense. We also show that a stronger statement is not true i.e. there are strongly one-sided dense $F _σ$ sets which are not $ {\mathrm{lip}}1$.

math.CA↗

arXiv2018

Type $1$ and $2$ sets for series of translates of functions

Suppose $Λ$ is a discrete infinite set of nonnegative real numbers. We say that $ Λ$ is type $1$ if the series $s(x)=\sum_{λ\inΛ}f(x+λ)$ satisfies a zero-one law. This means that for any non-negative measurable $f: {\mathbb R}\to [0,+ {\infty})$ either the convergence set $C(f, Λ)=\{x: s(x)<+ {\infty} \}= {\mathbb R}$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f, Λ)=\{x: s(x)=+ {\infty} \}= {\mathbb R}$ modulo sets of measure zero. If $ Λ$ is not type $1$ we say that $ Λ$ is type 2. The exact characterization of type $1$ and type $2$ sets is not known. In this paper we continue our study of the properties of type $1$ and $2$ sets. We discuss sub and supersets of type $1$ and $2$ sets and we give a complete and simple characterization of a subclass of dyadic type $1$ sets. We discuss the existence of type $1$ sets containing infinitely many algebraically independent elements. Finally, we consider unions and Minkowski sums of type $1$ and $2$ sets.

math.CA↗

arXiv2018

Random constructions for translates of non-negative functions

Suppose $Λ$ is a discrete infinite set of nonnegative real numbers. We say that $ Λ$ is type $2$ if the series $s(x)=\sum_{λ\inΛ}f(x+λ)$ does not satisfy a zero-one law. This means that we can find a non-negative measurable "witness function" $f: {\mathbb R}\to [0,+ {\infty})$ such that both the convergence set $C(f, Λ)=\{x: s(x)<+ {\infty} \}$ and its complement the divergence set $D(f, Λ)=\{x: s(x)=+ {\infty} \}$ are of positive Lebesgue measure. If $ Λ$ is not type $2$ we say that $ Λ$ is type $1$. The main result of our paper answers a question raised by Z. Buczolich, J-P. Kahane, and D. Mauldin. By a random construction we show that one can always choose a witness function which is the characteristic function of a measurable set. We also consider the effect on the type of a set $ Λ$ if we randomly delete its elements. Motivated by results concerning weighted sums $\sum c_n f(nx)$ and the Khinchin conjecture, we also discuss some results about weighted sums $\sum_{n=1}^{\infty}c_n f(x+λ_n)$.

math.CA↗

arXiv2018

On series of translates of positive functions III

Suppose $Λ$ is a discrete infinite set of nonnegative real numbers. We say that $ Λ$ is of type 1 if the series $s(x)=\sum_{λ\inΛ}f(x+λ)$ satisfies a zero-one law. This means that for any non-negative measurable $f: { {\mathbb R}}\to [0,+ {\infty})$ either the convergence set $C(f, Λ)=\{x: s(x)<+ {\infty} \}= { {\mathbb R}}$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f, Λ)=\{x: s(x)=+ {\infty} \}= { {\mathbb R}}$ modulo sets of measure zero. If $ Λ$ is not of type 1 we say that $ Λ$ is of type 2. In this paper we show that there is a universal $ Λ$ with gaps monotone decreasingly converging to zero such that for any open subset $G \subset { {\mathbb R}}$ one can find a characteristic function $f_{G}$ such that $G \subset D(f_G, Λ)$ and $C(f_G, Λ)= { {\mathbb R}} {\setminus} G$ modulo sets of measure zero. We also consider the question whether $C(f, Λ)$ can contain non-degenerate intervals for continuous functions when $D(f, Λ)$ is of positive measure. The above results answer some questions raised in a paper of Z. Buczolich, J-P. Kahane, and D. Mauldin.

math.CA↗