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Márton Elekes

Publications and source records attributed to Márton Elekes.

At least 37 records · Page 2Linked to original sources
arXiv2018

The structure of random homeomorphisms

In order to understand the structure of the "typical" element of a homeomorphism group, one has to study how large the conjugacy classes of the group are. When typical means generic in the sense of Baire category, this is well understood, see e.g. the works of Glasner and Weiss, and Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behaviour of the homeomorphisms is entirely different from the generic case. For $\text{Homeo}^+([0,1])$ we describe the non-Haar null conjugacy classes and also show that their union is co-Haar null, for $\text{Homeo}^+(\mathbb{S}^1)$ we describe the non-Haar null conjugacy classes, and for $\mathcal{U}(\ell^2)$ we show that, apart from the classes of the multishifts, all conjugacy classes are Haar null. As an application we affirmatively answer the question whether these groups can be written as the union of a meagre and a Haar null set.

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arXiv2018

The structure of random automorphisms of the rational numbers

In order to understand the structure of the "typical" element of an automorphism group, one has to study how large the conjugacy classes of the group are. For the case when typical is meant in the sense of Baire category, Truss proved that there is a co-meagre conjugacy class in Aut(Q, <), the automorphism group of the rational numbers. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. We give a complete description of the size of the conjugacy classes of Aut(Q, <) with respect to this notion. In particular, we show that there exist continuum many non-Haar null conjugacy classes, illustrating that the random behaviour is quite different from the typical one in the sense of Baire category.

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arXiv2016

Characterization of order types of pointwise linearly ordered families of Baire class 1 functions

In the 1970s M. Laczkovich posed the following problem: Let $\mathcal{B}_1(X)$ denote the set of Baire class $1$ functions defined on an uncountable Polish space $X$ equipped with the pointwise ordering. \[\text{Characterize the order types of the linearly ordered subsets of $\mathcal{B}_1(X)$.} \]The main result of the present paper is a complete solution to this problem. We prove that a linear order is isomorphic to a linearly ordered family of Baire class $1$ functions iff it is isomorphic to a subset of the following linear order that we call $([0,1]^{<ω_1}_{\searrow 0},<_{altlex})$, where $[0,1]^{<ω_1}_{\searrow 0}$ is the set of strictly decreasing transfinite sequences of reals in $[0, 1]$ with last element $0$, and $<_{altlex}$, the so called \emph{alternating lexicographical ordering}, is defined as follows: if $(x_α)_{α\leq ξ}, (x'_α)_{α\leq ξ'} \in [0,1]^{<ω_1}_{\searrow 0}$, and $δ$ is the minimal ordinal where the two sequences differ then we say that \[ (x_α)_{α\leq ξ} <_{altlex} (x'_α)_{α\leq ξ'} \iff (δ\text{ is even and } x_δ x'_δ). \] Using this characterization we easily reprove all the known results and answer all the known open questions of the topic.

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arXiv2016

Bruckner--Garg-type results with respect to Haar null sets in $C[0,1]$

A set $\mathcal{A}\subset C[0,1]$ is \emph{shy} or \emph{Haar null } (in the sense of Christensen) if there exists a Borel set $\mathcal{B}\subset C[0,1]$ and a Borel probability measure $μ$ on $C[0,1]$ such that $\mathcal{A}\subset \mathcal{B}$ and $μ\left(\mathcal{B}+f\right) = 0$ for all $f \in C[0,1]$. The complement of a shy set is called a \emph{prevalent} set. We say that a set is \emph{Haar ambivalent} if it is neither shy nor prevalent. The main goal of the paper is to answer the following question: What can we say about the topological properties of the level sets of the prevalent/non-shy many $f\in C[0,1]$? The classical Bruckner--Garg Theorem characterizes the level sets of the generic (in the sense of Baire category) $f\in C[0,1]$ from the topological point of view. We prove that the functions $f\in C[0,1]$ for which the same characterization holds form a Haar ambivalent set. In an earlier paper we proved that the functions $f\in C[0,1]$ for which positively many level sets with respect to the Lebesgue measure $λ$ are singletons form a non-shy set in $C[0,1]$. The above result yields that this set is actually Haar ambivalent. Now we prove that the functions $f\in C[0,1]$ for which positively many level sets with respect to the occupation measure $λ\circ f^{-1}$ are not perfect form a Haar ambivalent set in $C[0,1]$. We show that for the prevalent $f\in C[0,1]$ for the generic $y\in f([0,1])$ the level set $f^{-1}(y)$ is perfect. Finally, we answer a question of Darji and White by showing that the set of functions $f \in C[0,1]$ for which there exists a perfect $P_f\subset [0,1]$ such that $f'(x) = \infty$ for all $x \in P_f$ is Haar ambivalent.

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arXiv2015

Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps

The notions of shyness and prevalence generalize the property of being zero and full Haar measure to arbitrary (not necessarily locally compact) Polish groups. The main goal of the paper is to answer the following question: What can we say about the Hausdorff and packing dimension of the fibers of prevalent continuous maps? Let $K$ be an uncountable compact metric space. We prove that the prevalent $f\in C(K,\mathbb{R}^d)$ has many fibers with almost maximal Hausdorff dimension. This generalizes a theorem of Dougherty and yields that the prevalent $f\in C(K,\mathbb{R}^d)$ has graph of maximal Hausdorff dimension, generalizing a result of Bayart and Heurteaux. We obtain similar results for the packing dimension. We show that for the prevalent $f\in C([0,1]^m,\mathbb{R}^d)$ the set of $y\in f([0,1]^m)$ for which $\dim_H f^{-1}(y)=m$ contains a dense open set having full measure with respect to the occupation measure $λ^m \circ f^{-1}$, where $\dim_H$ and $λ^m$ denote the Hausdorff dimension and the $m$-dimensional Lebesgue measure, respectively. We also prove an analogous result when $[0,1]^m$ is replaced by any self-similar set satisfying the open set condition. We cannot replace the occupation measure with Lebesgue measure in the above statement: We show that the functions $f\in C[0,1]$ for which positively many level sets are singletons form a non-shy set in $C[0,1]$. In order to do so, we generalize a theorem of Antunović, Burdzy, Peres and Ruscher. As a complementary result we prove that the functions $f\in C[0,1]$ for which $\dim_H f^{-1}(y)=1$ for all $y\in (\min f,\max f)$ form a non-shy set in $C[0,1]$. We also prove sharper results in which large Hausdorff dimension is replaced by positive measure with respect to generalized Hausdorff measures, which answers a problem of Fraser and Hyde.

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arXiv2015

Naively Haar null sets in Polish groups

Let $(G,\cdot)$ be a Polish group. We say that a set $X \subset G$ is Haar null if there exists a universally measurable set $U \supset X$ and a Borel probability measure $μ$ such that for every $g, h \in G$ we have $μ(gUh)=0$. We call a set $X$ naively Haar null if there exists a Borel probability measure $μ$ such that for every $g, h \in G$ we have $μ(gXh)=0$. Generalizing a result of Elekes and Steprāns, which answers the first part of Problem FC from Fremlin's list, we prove that in every abelian Polish group there exists a naively Haar null set that is not Haar null.

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arXiv2015

Ranks on the Baire class $ξ$ functions

In 1990 Kechris and Louveau developed the theory of three very natural ranks on the Baire class $1$ functions. A rank is a function assigning countable ordinals to certain objects, typically measuring their complexity. We extend this theory to the case of Baire class $ξ$ functions, and generalize most of the results from the Baire class 1 case. We also show that their assumption of the compactness of the underlying space can be eliminated. As an application, we solve a problem concerning the so called solvability cardinals of systems of difference equations, arising from the theory of geometric decompositions. We also show that certain other very natural generalizations of the ranks of Kechris and Louveau surprisingly turn out to be bounded in $ω_1$. Finally, we prove a general result showing that all ranks satisfying some natural properties coincide for bounded functions.

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arXiv2015

A new fractal dimension: The topological Hausdorff dimension

We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension. We examine the basic properties of this new notion of dimension, compare it to other well-known notions, determine its value for some classical fractals such as the Sierpinski carpet, the von Koch snowflake curve, Kakeya sets, the trail of the Brownian motion, etc. As our first application, we generalize the celebrated result of Chayes, Chayes and Durrett about the phase transition of the connectedness of the limit set of Mandelbrot's fractal percolation process. They proved that certain curves show up in the limit set when passing a critical probability, and we prove that actually `thick' families of curves show up, where roughly speaking the word thick means that the curves can be parametrized in a natural way by a set of large Hausdorff dimension. The proof of this is basically a lower estimate of the topological Hausdorff dimension of the limit set. For the sake of completeness, we also give an upper estimate and conclude that in the non-trivial cases the topological Hausdorff dimension is almost surely strictly below the Hausdorff dimension. Finally, as our second application, we show that the topological Hausdorff dimension is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic continuous function (in the sense of Baire category) defined on a compact metric space.

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arXiv2014

Decomposing the real line into Borel sets closed under addition

We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in $ZFC$ and even in the theory $ZFC + \mathfrak{c} = ω_2$ if the number of pieces can be uncountable but less than the continuum. We also investigate various versions: what happens if we drop the Borelness requirement, if we replace addition by multiplication, if the pieces are subgroups, if we partition $(0,\infty)$, and so on.

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arXiv2014

Haar null sets without $G_δ$ hulls

Let $G$ be an abelian Polish group, e.g. a separable Banach space. A subset $X \subset G$ is called Haar null (in the sense of Christensen) if there exists a Borel set $B \supset X$ and a Borel probability measure $μ$ on $G$ such that $μ(B+g)=0$ for every $g \in G$. The term shy is also commonly used for Haar null, and co-Haar null sets are often called prevalent. Answering an old question of Mycielski we show that if $G$ is not locally compact then there exists a Borel Haar null set that is not contained in any $G_δ$ Haar null set. We also show that $G_δ$ can be replaced by any other class of the Borel hierarchy, which implies that the additivity of the $σ$-ideal of Haar null sets is $ω_1$. The definition of a generalised Haar null set is obtained by replacing the Borelness of $B$ in the above definition by universal measurability. We give an example of a generalised Haar null set that is not Haar null, more precisely we construct a coanalytic generalised Haar null set without a Borel Haar null hull. This solves Problem GP from Fremlin's problem list. Actually, all our results readily generalise to all Polish groups that admit a two-sided invariant metric.

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arXiv2013

Haar null sets and the consistent reflection of non-meagreness

A subset $X$ of a Polish group $G$ is called \emph{Haar null} if there exists a Borel set $B \supset X$ and Borel probability measure $μ$ on $G$ such that $μ(gBh)=0$ for every $g,h \in G$. We prove that there exists a set $X \subset \mathbb{R}$ that is not Lebesgue null and a Borel probability measure $μ$ such that $μ(X + t) = 0$ for every $t \in \mathbb{R}$. This answers a question from David Fremlin's problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.) This result motivates the following Baire category analogue. It is consistent with $ZFC$ that there exist an abelian Polish group $G$ and a Cantor set $C \subset G$ such that for every non-meagre set $X \subset G$ there exists a $t \in G$ such that $C \cap (X + t)$ is relatively non-meagre in $C$. This essentially generalises results of Bartoszyński and Burke-Miller.

math.CA↗
arXiv2013

Answer to a question of Kolmogorov

More than 80 years ago Kolmogorov asked the following question. Let $E\subseteq \mathbb{R}^{2}$ be a measurable set with $λ^{2}(E)<\infty$, where $λ^2$ denotes the two-dimensional Lebesgue measure. Does there exist for every $\varepsilon>0$ a contraction $f\colon E\to \mathbb{R}^{2}$ such that $λ^{2}(f(E))\geq λ^{2}(E)-\varepsilon$ and $f(E)$ is a polygon? We answer this question in the negative by constructing a bounded, simply connected open counterexample. Our construction can easily be modified to yield the analogous result in higher dimensions.

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arXiv2012

Reconstructing geometric objects from the measures of their intersections with test sets

Let us say that an element of a given family $\A$ of subsets of $\R^d$ can be reconstructed using $n$ test sets if there exist $T_1,...,T_n \subset \R^d$ such that whenever $A,B\in \A$ and the Lebesgue measures of $A \cap T_i$ and $B \cap T_i$ agree for each $i=1,...,n$ then $A=B$. Our goal will be to find the least such $n$. We prove that if $\A$ consists of the translates of a fixed reasonably nice subset of $\R^d$ then this minimum is $n=d$. In order to obtain this result we reconstruct a translate of a fixed function using $d$ test sets as well, and also prove that under rather mild conditions the measure function $f_{K,θ} (r) = \la^{d-1} (K \cap \{x \in \RR^d : = r\})$ of the sections of $K$ is absolutely continuous for almost every direction $θ$. These proofs are based on techniques of harmonic analysis. We also show that if $\A$ consists of the magnified copies $rE+t$ $(r\ge 1, t\in\R^d)$ of a fixed reasonably nice set $E\subset \R^d$, where $d\ge 2$, then $d+1$ test sets reconstruct an element of $\A$. This fails in $\R$: we prove that an interval, and even an interval of length at least 1 cannot be reconstructed using 2 test sets. Finally, using randomly constructed test sets, we prove that an element of a reasonably nice $k$-dimensional family of geometric objects can be reconstructed using $2k+1$ test sets. A example from algebraic topology shows that $2k+1$ is sharp in general.

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arXiv2011

Chains of Baire class 1 functions and various notions of special trees

Following Laczkovich we consider the partially ordered set $\iB_1(\RR)$ of Baire class 1 functions endowed with the pointwise order, and investigate the order types of the linearly ordered subsets. Answering a question of Komjáth and Kunen we show (in $ZFC$) that special Aronszajn lines are embeddable into $\iB_1(\RR)$. We also show that under Martin's Axiom a linearly ordered set $\mathbb{L}$ with $|\mathbb{L}|<2^ω$ is embeddable into $\iB_1(\RR)$ iff $\mathbb{L}$ does not contain a copy of $ω_1$ or $ω_1^*$. We present a $ZFC$-example of a linear order of size $2^ω$ showing that this characterisation is not valid for orders of size continuum. These results are obtained using the notion of a compact-special tree; that is, a tree that is embeddable into the class of compact subsets of the reals partially ordered under reverse inclusion. We investigate how this notion is related to the well-known notion of an $\RR$-special tree and also to some other notions of specialness.

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arXiv2011

Transfinite Sequences of Continuous and Baire Class 1 Functions

The set of continuous or Baire class 1 functions defined on a metric space $X$ is endowed with the natural pointwise partial order. We investigate how the possible lengths of well-ordered monotone sequences (with respect to this order) depend on the space $X$.

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arXiv2011

The Fixed Point of the Composition of Derivatives

We give an affirmative answer to a question of K. Ciesielski by showing that the composition $f\circ g$ of two derivatives $f,g:[0,1]\to[0,1]$ always has a fixed point. Using Maximoff's Theorem we obtain that the composition of two $[0,1]\to[0,1]$ Darboux Baire-1 functions must also have a fixed point.

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arXiv2011

Measurable Envelopes, Hausdorff Measures and Sierpiński Sets

We show that the existence of measurable envelopes of all subsets of $\RR^n$ with respect to the $d$-dimensional Hausdorff measure $(0<d<n)$ is independent of $ZFC$. We also investigate the consistency of the existence of Sierpiński sets measurable with respect to the $d$-dimensional Hausdorff measure.

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