arXiv subjects

Tamás Keleti

Publications and source records attributed to Tamás Keleti.

At least 19 recordsLinked to original sources

arXiv2026

New Hausdorff type dimensions and optimal bounds for bilipschitz invariant dimensions

We introduce a new family of fractal dimensions by restricting the set of diameters in the coverings in the usual definition of the Hausdorff dimension. Among others, we prove that this family contains continuum many distinct dimensions, and they share most of the properties of the Hausdorff dimension, which answers negatively a question of Fraser. On the other hand, we also prove that among these new dimensions only the Hausdorff dimension behaves nicely with respect to Hölder functions. We also consider the supremum of these new dimensions, which turns out to be another interesting notion of fractal dimension. We prove that among those bilipschitz invariant, monotone dimensions on the compact subsets of $\mathbb{R}^n$ that agree with the similarity dimension for the simplest self-similar sets, the modified lower dimension is the smallest and when $n=1$ the Assouad dimension is the greatest, and this latter statement is false for $n>1$. This answers a question of Rutar.

math.CA↗

arXiv2025

Generic properties of topological groups

We study generic properties of topological groups in the sense of Baire category. First we investigate countably infinite (discrete) groups. We extend a classical result of B. H. Neumann, H. Simmons and A. Macintyre on algebraically closed groups and the word problem. I. Goldbring, S. E. Kunnawalkam and Y. Lodha proved that every isomorphism class is meager among countably infinite (discrete) groups. In contrast, we show that there is a comeager isomorphism class among countably infinite (discrete) abelian groups. Then we turn to compact metrizable abelian groups. We use Pontryagin duality to show that there is a comeager isomorphism class among compact metrizable abelian groups. We discuss its connections to the countably infinite (discrete) case. Finally, we study compact metrizable groups. We prove that the generic compact metrizable group is neither connected nor totally disconnected; also it is neither torsion-free nor a torsion group.

math.LO↗

arXiv2024

Lipschitz images and dimensions

We consider the question which compact metric spaces can be obtained as a Lipschitz image of the middle third Cantor set, or more generally, as a Lipschitz image of a subset of a given compact metric space. In the general case we prove that if $A$ and $B$ are compact metric spaces and the Hausdorff dimension of $A$ is bigger than the upper box dimension of $B$, then there exist a compact set $A'\subset A$ and a Lipschitz onto map $f\colon A'\to B$. As a corollary we prove that any `natural' dimension in $\mathbb{R}^n$ must be between the Hausdorff and upper box dimensions. We show that if $A$ and $B$ are self-similar sets with the strong separation condition with equal Hausdorff dimension and $A$ is homogeneous, then $A$ can be mapped onto $B$ by a Lipschitz map if and only if $A$ and $B$ are bilipschitz equivalent. For given $α>0$ we also give a characterization of those compact metric spaces that can be obtained as an $α$-Hölder image of a compact subset of $\mathbb{R}$. The quantity we introduce for this turns out to be closely related to the upper box dimension.

math.CA↗

arXiv2024

Hausdorff dimension of union of lines that cover a curve

We construct a continuously differentiable curve in the plane that can be covered by a collection of lines such that every line intersects the curve at a single point and the union of the lines has Hausdorff dimension 1. We show that for twice differentiable curves this is impossible. In that case, the union of the lines must have Hausdorff dimension 2. If we use only tangent lines then the differentiability of the curve already implies that the union of the lines must have Hausdorff dimension 2, unless the curve is a line. We also construct a continuous curve, which is in fact the graph of a strictly convex function, such that the union of (one sided) tangent lines has Hausdorff dimension 1.

math.MG↗

arXiv2023

Equivalences between different forms of the Kakeya conjecture and duality of Hausdorff and packing dimensions for additive complements

The Kakeya conjecture is generally formulated as one the following statements: every compact/Borel/arbitrary subset of ${\mathbb R}^n$ that contains a (unit) line segment in every direction has Hausdorff dimension $n$; or, sometimes, that every closed/Borel/arbitrary subset of ${\mathbb R}^n$ that contains a full line in every direction has Hausdorff dimension $n$. These statements are generally expected to be equivalent. Moreover, the condition that the set contains a line (segment) in every direction is often relaxed by requiring a line (segment) for a "large" set of directions only, where large could mean a set of positive $(n-1)$-dimensional Lebesgue measure. Here we prove that all the above forms of the Kakeya conjecture are indeed equivalent. In fact, we prove that there exist $d\le n$ and a compact subset $C$ of ${\mathbb R}^n$ of Hausdorff dimension $d$ that contains a unit line segment in every direction (and also a closed set of dimension $d$ that contains a line in every direction) such that every subset $S$ of ${\mathbb R}^n$ that contains a line segment in every direction of a set of Hausdorff dimension $n-1$, must have dimension at least $d$. We also obtain results on the duality of Hausdorff and packing dimensions via additive complements: For any non-empty Borel set $A$ of ${\mathbb R}^n$ we show that (1) the Hausdorff dimension of $A$ can be obtained as $n-p$, where $p$ is the infimum of the packing dimension of those Borel subsets $B$ of ${\mathbb R}^n$ for which $A+B={\mathbb R}^n$; and (2) the packing dimension of $A$ can be obtained as $n-h$, where $h$ is the infimum of the Hausdorff dimension of those Borel subsets $B$ of ${\mathbb R}^n$ for which $A+B={\mathbb R}^n$.

math.MG↗

arXiv2022

Tiling of rectangles with squares via Diophantine approximation

This article shines new light on the classical problem of tiling rectangles with squares efficiently with a novel method. With a twist on the traditional approach of resistor networks, we provide new and improved results on the matter using the theory of Diophantine Approximation, hence overcoming long-established difficulties, such as generalizations to higher-dimensional analogues. The universality of the method is demonstrated through its applications to different tiling problems. These include tiling rectangles with other rectangles, with their respective higher-dimensional counterparts, as well as tiling equilateral triangles, parallelograms, and trapezoids with equilateral triangles.

math.CO↗

arXiv2022

Large sets avoiding infinite arithmetic / geometric progressions

We study some variants of the Erdős similarity problem. We pose the question if every measurable subset of the real line with positive measure contains a similar copy of an infinite geometric progression. We construct a compact subset $E$ of the real line such that $0$ is a Lebesgue density point of $E$, but $E$ does not contain any (non-constant) infinite geometric progression. We give a sufficient density type condition that guarantees that a set contains an infinite geometric progression. By slightly improving a recent result of Bradford, Kohut and Mooroogen arXiv:2205.04786, we construct a closed set $F\subset[0,\infty)$ such that the measure of $F\cap[t,t+1]$ tends to $1$ at infinity but $F$ does not contain any infinite arithmetic progression. We also slightly improve a more general recent result by Kolountzakis and Papageorgiou arXiv:2208.02637 for more general sequences. We give a sufficient condition that guarantees that a given Cantor type set contains at least one infinite geometric progression with any quotient between $0$ and $1$. This can be applied to most symmetric Cantor sets of positive measure.

math.MG↗

arXiv2021

Generic countably infinite groups

Countably infinite groups (with a fixed underlying set) constitute a Polish space $G$ with a suitable metric, hence the Baire category theorem holds in $G$. We study isomorphism invariant subsets of $G$, which we call group properties. We say that the generic countably infinite group is of property $P$ if $P$ is comeager in $G$. We prove that every group property with the Baire property is either meager or comeager. We show that there is a comeager elementary equivalence class in $G$ but every isomorphism class is meager. We prove that the generic group is algebraically closed, simple, not finitely generated and not locally finite. We show that in the subspace of Abelian groups the generic group is isomorphic to the unique countable, divisible torsion group that contains every finite Abelian group. We sketch the model-theoretic setting in which many of our results can be generalized. We briefly discuss a connection with infinite games.

math.LO↗

arXiv2021

Fractal percolation is unrectifiable

We show that there exists $0<α_0<1$ (depending on the parameters) such that the fractal percolation is almost surely purely $α$-unrectifiable for all $α>α_0$.

math.PR↗

arXiv2020

Large Sets with Small Injective Projections

Let $\ell_1,\ell_2,\dots$ be a countable collection of lines in ${\mathbb R}^d$. For any $t \in [0,1]$ we construct a compact set $Γ\subset{\mathbb R}^d$ with Hausdorff dimension $d-1+t$ which projects injectively into each $\ell_i$, such that the image of each projection has dimension $t$. This immediately implies the existence of homeomorphisms between certain Cantor-type sets whose graphs have large dimensions. As an application, we construct a collection $E$ of disjoint, non-parallel $k$-planes in $\mathbb{R}^d$, for $d \geq k+2$, whose union is a small subset of $\mathbb{R}^d$, either in Hausdorff dimension or Lebesgue measure, while $E$ itself has large dimension. As a second application, for any countable collection of vertical lines $w_i$ in the plane we construct a collection of nonvertical lines $H$, so that $F$, the union of lines in $H$, has positive Lebesgue measure, but each point of each line $w_i$ intersects at most one $h\in H$ and, for each $w_i$, the Hausdorff dimension of $F\cap w_i$ is zero.

math.MG↗

arXiv2019

New bounds on the dimensions of planar distance sets

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if $A\subset\mathbb{R}^2$ is a Borel set of Hausdorff dimension $s>1$, then its distance set has Hausdorff dimension at least $37/54\approx 0.685$. Moreover, if $s\in (1,3/2]$, then outside of a set of exceptional $y$ of Hausdorff dimension at most $1$, the pinned distance set $\{ |x-y|:x\in A\}$ has Hausdorff dimension $\ge \tfrac{2}{3}s$ and packing dimension at least $ \tfrac{1}{4}(1+s+\sqrt{3s(2-s)}) \ge 0.933$. These estimates improve upon the existing ones by Bourgain, Wolff, Peres-Schlag and Iosevich-Liu for sets of Hausdorff dimension $>1$. Our proof uses a multi-scale decomposition of measures in which, unlike previous works, we are able to choose the scales subject to certain constrains. This leads to a combinatorial problem, which is a key new ingredient of our approach, and which we solve completely by optimizing certain variation of Lipschitz functions.

math.CA↗

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↗

arXiv2017

Small union with large set of centers

Let $T\subset{\mathbb R}^n$ be a fixed set. By a scaled copy of $T$ around $x\in{\mathbb R}^n$ we mean a set of the form $x+rT$ for some $r>0$. In this survey paper we study results about the following type of problems: How small can a set be if it contains a scaled copy of $T$ around every point of a set of given size? We will consider the cases when $T$ is circle or sphere centered at the origin, Cantor set in ${\mathbb R}$, the boundary of a square centered at the origin, or more generally the $k$-skeleton ($0\le k<n$) of an $n$-dimensional cube centered at the origin or the $k$-skeleton of a more general polytope of ${\mathbb R}^n$. We also study the case when we allow not only scaled copies but also scaled and rotated copies and also the case when we allow only rotated copies.

math.MG↗

arXiv2015

Better bounds for planar sets avoiding unit distances

A $1$-avoiding set is a subset of $\mathbb{R}^n$ that does not contain pairs of points at distance $1$. Let $m_1(\mathbb{R}^n)$ denote the maximum fraction of $\mathbb{R}^n$ that can be covered by a measurable $1$-avoiding set. We prove two results. First, we show that any $1$-avoiding set in $\mathbb{R}^n$ ($n\ge 2$) that displays block structure (i.e., is made up of blocks such that the distance between any two points from the same block is less than $1$ and points from distinct blocks lie farther than $1$ unit of distance apart from each other) has density strictly less than $1/2^n$. For the special case of sets with block structure this proves a conjecture of Erdős asserting that $m_1(\mathbb{R}^2) < 1/4$. Second, we use linear programming and harmonic analysis to show that $m_1(\mathbb{R}^2) \leq 0.258795$.

math.MG↗

arXiv2015

Are lines much bigger than line segments?

We pose the following conjecture: (*) If A is the union of line segments in R^n, and B is the union of the corresponding full lines then the Hausdorff dimensions of A and B agree. We prove that this conjecture would imply that every Besicovitch set (compact set that contains line segments in every direction) in R^n has Hausdorff dimension at least n-1 and (upper) Minkowski dimension n. We also prove that conjecture (*) holds if the Hausdorff dimension of B is at most 2, so in particular it holds in the plane.

math.MG↗

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.

math.LO↗

arXiv2014

Squares and their centers

We study the relationship between the sizes of two sets $B, S\subset\mathbb{R}^2$ when $B$ contains either the whole boundary, or the four vertices, of a square with axes-parallel sides and center in every point of $S$, where size refers to one of cardinality, Hausdorff dimension, packing dimension, or upper or lower box dimension. Perhaps surprinsingly, the results vary depending on the notion of size under consideration. For example, we construct a compact set $B$ of Hausdorff dimension $1$ which contains the boundary of an axes-parallel square with center in every point $[0,1]^2$, but prove that such a $B$ must have packing and lower box dimension at least $\tfrac{7}{4}$, and show by example that this is sharp. For more general sets of centers, the answers for packing and box counting dimensions also differ. These problems are inspired by the analogous problems for circles that were investigated by Bourgain, Marstrand and Wolff, among others.

math.MG↗

arXiv2012

Hausdorff dimension of metric spaces and Lipschitz maps onto cubes

We prove that a compact metric space (or more generally an analytic subset of a complete separable metric space) of Hausdorff dimension bigger than $k$ can be always mapped onto a $k$-dimensional cube by a Lipschitz map. We also show that this does not hold for arbitrary separable metric spaces. As an application we essentially answer a question of Urbański by showing that the transfinite Hausdorff dimension (introduced by him) of an analytic subset $A$ of a complete separable metric space is the integer part of $\dim_H A$ if $\dim_H A$ is finite but not an integer, $\dim_H A$ or $\dim_H A-1$ if $\dim_H A$ is an integer and at least $ω_0$ if $\dim_H A=\infty$.

math.CA↗