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András Máthé

Publications and source records attributed to András Máthé.

At least 19 recordsLinked to original sources
arXiv2026

Moser-Tardos Algorithm with small number of random bits

We study a variant of the parallel Moser-Tardos Algorithm. We prove that if we restrict attention to a class of problems whose dependency graphs have subexponential growth, then the expected total number of random bits used by the algorithm is constant; in particular, it is independent from the number of variables. This is achieved by using the same random bits to resample variables which are far enough in the dependency graph. There are two corollaries. First, we obtain a deterministic algorithm for finding a satisfying assignment, which for any class of problems as in the previous paragraph runs in time O(n), where n is the number of variables. Second, we present a Borel version of the Lovász Local Lemma.

math.CO↗
arXiv2025

Circle Squaring with Pieces of Small Boundary and Low Borel Complexity

Tarski's Circle Squaring Problem from 1925 asks whether it is possible to partition a disk in the plane into finitely many pieces and reassemble them via isometries to yield a partition of a square of the same area. It was finally resolved by Laczkovich in 1990 in the affirmative. Recently, several new proofs have emerged which achieve circle squaring with better structured pieces: namely, pieces which are Lebesgue measurable and have the property of Baire (Grabowski-Máthé-Pikhurko) or even are Borel (Marks-Unger). In this paper, we show that circle squaring is possible with Borel pieces of positive Lebesgue measure whose boundaries have upper Minkowski dimension less than 2 (in particular, each piece is Jordan measurable). We also improve the Borel complexity of the pieces: namely, we show that each piece can be taken to be a Boolean combination of $F_σ$ sets. This is a consequence of our more general result that applies to any two bounded subsets of $R^k$, $k\ge 1$, of equal positive measure whose boundaries have upper Minkowski dimension smaller than $k$.

math.MG↗
arXiv2025

Discretised sum-product theorems by Shannon-type inequalities

By making use of arithmetic information inequalities, we give a strong quantitative bound for the discretised ring theorem. In particular, we show that if $A \subset [1,2]$ is a $(δ,σ)$-set, with $|A| = δ^{-σ},$ then $A+A$ or $AA$ has $δ$-covering number at least $δ^{-c}|A|$ for any $0 < c < \min\{σ/6, (1-σ)/6\}$ provided that $δ> 0$ is small enough.

math.CA↗
arXiv2024

Borel version of the Local Lemma

We prove a Borel version of the local lemma, i.e. we show that, under suitable assumptions, if the set of variables in the local lemma has a structure of a Borel space, then there exists a satisfying assignment which is a Borel function. The main tool which we develop for the proof, which is of independent interest, is a parallel version of the Moser-Tardos algorithm which uses the same random bits to resample clauses that are far enough in the dependency graph.

math.CO↗
arXiv2023

Measurable equidecompositions for group actions with an expansion property

Given an action of a group $Γ$ on a measure space $Ω$, we provide a sufficient criterion under which two sets $A, B\subseteq Ω$ are measurably equidecomposable, i.e., $A$ can be partitioned into finitely many measurable pieces which can be rearranged using the elements of $Γ$ to form a partition of $B$. In particular, we prove that every bounded measurable subset of $R^n$, $n\ge 3$, with non-empty interior is measurably equidecomposable to a ball via isometries. The analogous result also holds for some other spaces, such as the sphere or the hyperbolic space of dimension $n\ge 2$.

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↗
arXiv2016

Restrictions of Hölder continuous functions

For $0<α<1$ let $V(α)$ denote the supremum of the numbers $v$ such that every $α$-Hölder continuous function is of bounded variation on a set of Hausdorff dimension $v$. Kahane and Katznelson (2009) proved the estimate $1/2 \leq V(α)\leq 1/(2-α)$ and asked whether the upper bound is sharp. We show that in fact $V(α)=\max\{1/2,α\}$. Let $\dim_{H}$ and $\overline{\dim}_{M}$ denote the Hausdorff and upper Minkowski dimension, respectively. The upper bound on $V(α)$ is a consequence of the following theorem. Let $\{B(t): t\in [0,1]\}$ be a fractional Brownian motion of Hurst index $α$. Then, almost surely, there exists no set $A\subset [0,1]$ such that $\overline{\dim}_{M} A>\max\{1-α,α\}$ and $B\colon A\to \mathbb{R}$ is of bounded variation. Furthermore, almost surely, there exists no set $A\subset [0,1]$ such that $\overline{\dim}_{M} A>1-α$ and $B\colon A\to \mathbb{R}$ is $β$-Hölder continuous for some $β>α$. The zero set and the set of record times of $B$ witness that the above theorems give the optimal dimensions. We also prove similar restriction theorems for deterministic self-affine functions and generic $α$-Hölder continuous functions. Finally, let $\{\mathbf{B}(t): t\in [0,1]\}$ be a two-dimensional Brownian motion. We prove that, almost surely, there is a compact set $D\subset [0,1]$ such that $\dim_{H} D\geq 1/3$ and $\mathbf{B}\colon D\to \mathbb{R}^2$ is non-decreasing in each coordinate. It remains open whether $1/3$ is best possible.

math.PR↗
arXiv2016

Cliques in dense inhomogeneous random graphs

The theory of dense graph limits comes with a natural sampling process which yields an inhomogeneous variant G(n,W) of the Erdos-Renyi random graph. Here we study the clique number of these random graphs. We establish the concentration of the clique number of G(n,W) for each fixed n, and give examples of graphons for which G(n,W) exhibits wild long-term behavior. Our main result is an asymptotic formula which gives the almost sure clique number of these random graphs. We obtain a similar result for the bipartite version of the problem. We also make an observation that might be of independent interest: Every graphon avoiding a fixed graph is countably-partite.

math.CO↗
arXiv2016

Measurable circle squaring

Laczkovich proved that if bounded subsets $A$ and $B$ of $R^k$ have the same non-zero Lebesgue measure and the box dimension of the boundary of each set is less than $k$, then there is a partition of $A$ into finitely many parts that can be translated to form a partition of $B$. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, this gives measurable and translation-only versions of Tarski's circle squaring and Hilbert's third problem.

math.MG↗
arXiv2015

Inhomogeneous self-similar sets with overlaps

It is known that if the underlying iterated function system satisfies the open set condition, then the upper box dimension of an inhomogeneous self-similar set is the maximum of the upper box dimensions of the homogeneous counterpart and the condensation set. First, we prove that this `expected formula' does not hold in general if there are overlaps in the construction. We demonstrate this via two different types of counterexample: the first is a family of overlapping inhomogeneous self-similar sets based upon Bernoulli convolutions; and the second applies in higher dimensions and makes use of a spectral gap property that holds for certain subgroups of $SO(d)$ for $d\geq 3$. We also obtain new upper bounds for the upper box dimension of an inhomogeneous self-similar set which hold in general. Moreover, our counterexamples demonstrate that these bounds are optimal. In the final section we show that if the \emph{weak separation property} is satisfied, ie. the overlaps are controllable, then the `expected formula' does hold.

math.CA↗
arXiv2015

Measuring sets with translation invariant Borel measures

Following Davies, Elekes and Keleti, we study measured sets, i.e. Borel sets $B$ in $\mathbb{R}$ (or in a Polish group) for which there is a translation invariant Borel measure assigning positive and σ-finite measure to $B$. We investigate which sets can be written as a (disjoint) union of measured sets. We show that every Borel nullset $B\subset \mathbb{R}$ of the second category is larger than any nullset $A\subset \mathbb{R}$ in the sense that there are partitions $B=B_1\cup B_2$, $A=A_1\cup A_2$ and gauge functions $g_1, g_2$ such that the Hausdorff measures satisfy $H^{g_i}(B_i)=1$ and $H^{g_i}(A_i)=0$ ($i=1,2$). This implies that every Borel set of the second category is a union of two measured sets. We also present Borel and compact sets in $\mathbb{R}$ which are not a union of countably many measured sets. This is done in two steps. First we show that non-locally compact Polish groups are not a union of countably many measured sets. Then, to certain Banach spaces we associate a Borel and/or σ-compact additive subgroup of $\mathbb{R}$ which is not a union of countably many measured sets. It is also shown that there are measured sets which are null or non-σ-finite for every Hausdorff measure of arbitrary gauge function.

math.FA↗
arXiv2014

Hamilton cycles in dense vertex-transitive graphs

A famous conjecture of Lovász states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and sufficiently large. In fact, we show that such graphs contain a Hamilton cycle and moreover we provide a polynomial time algorithm for finding such a cycle.

math.CO↗
arXiv2013

Generalized Hausdorff measure for generic compact sets

Let $X$ be a Polish space. We prove that the generic compact set $K\subseteq X$ (in the sense of Baire category) is either finite or there is a continuous gauge function $h$ such that $0<\mathcal{H}^{h}(K)<\infty$, where $\mathcal{H}^h$ denotes the $h$-Hausdorff measure. This answers a question of C. Cabrelli, U. B. Darji, and U. M. Molter. Moreover, for every weak contraction $f\colon K\to X$ we have $\mathcal{H}^{h} (K\cap f(K))=0$. This is a measure theoretic analogue of a result of M. Elekes.

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.

math.CA↗
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↗
arXiv2012

How large dimension guarantees a given angle?

We study the following two problems: (1) Given $n\ge 2$ and $\al$, how large Hausdorff dimension can a compact set $A\su\Rn$ have if $A$ does not contain three points that form an angle $\al$? (2) Given $\al$ and $\de$, how large Hausdorff dimension can a %compact subset $A$ of a Euclidean space have if $A$ does not contain three points that form an angle in the $\de$-neighborhood of $\al$? An interesting phenomenon is that different angles show different behaviour in the above problems. Apart from the clearly special extreme angles 0 and $180^\circ$, the angles $60^\circ,90^\circ$ and $120^\circ$ also play special role in problem (2): the maximal dimension is smaller for these special angles than for the other angles. In problem (1) the angle $90^\circ$ seems to behave differently from other angles.

math.CA↗
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.

math.CA↗