arXiv subjects

András Máthé

Publications and source records attributed to András Máthé.

21 records · Page 2Linked to original sources

arXiv2012

Sets of large dimension not containing polynomial configurations

The main result of this paper is the following. Given countably many multivariate polynomials with rational coefficients and maximum degree $d$, we construct a compact set $E\subset \R^n$ of Hausdorff dimension $n/d$ which does not contain finite point configurations corresponding to the zero sets of the given polynomials. Given a set $E\subset \R^n$, we study the angles determined by three points of $E$. The main result implies the existence of a compact set in $\R^n$ of Hausdorff dimension $n/2$ which does not contain the angle $π/2$. (This is known to be sharp if $n$ is even.) We show that there is a compact set of Hausdorff dimension $n/8$ which does not contain an angle in any given countable set. We also construct a compact set $E\subset \R^n$ of Hausdorff dimension $n/6$ for which the set of angles determined by $E$ is Lebesgue null. In the other direction, we present a result that every set of sufficiently large dimension contains an angle $ε$ close to any given angle.

math.CA↗

arXiv2011

Can we assign the Borel hulls in a monotone way?

A \emph{hull} of $A \subset [0,1]$ is a set $H$ containing $A$ such that $λ^*(H)=λ^*(A)$. We investigate all four versions of the following problem. Does there exist a monotone (wrt. inclusion) map that assigns a Borel/$G_δ$ hull to every negligible/measurable subset of $[0,1]$? Three versions turn out to be independent of ZFC (the usual Zermelo-Fraenkel axioms with the Axiom of Choice), while in the fourth case we only prove that the nonexistence of a monotone $G_δ$ hull operation for all measurable sets is consistent. It remains open whether existence here is also consistent. We also answer a question of Z. Gyenes and D. Pálvölgyi which asks if monotone hulls can be defined for every chain (wrt. inclusion) of measurable sets. We also comment on the problem of hulls of all subsets of $[0,1]$.

math.CA↗

arXiv2008

Self-similar and self-affine sets; measure of the intersection of two copies

Let K be a self-similar or self-affine set in R^d, let μbe a self-similar or self-affine measure on it, and let G be the group of affine maps, similitudes, isometries or translations of R^d. Under various assumptions (such as separation conditions or we assume that the transformations are small perturbations or that K is a so called Sierpinski sponge) we prove theorems of the following types, which are closely related to each other; Non-stability: There exists a constant c<1 such that for every g\in G we have either μ(K\cap g(K)) 0 \iff int_K (K\cap g(K)) is nonempty (where int_K is interior relative to K). Extension: The measure μhas a G-invariant extension to R^d. Moreover, in many situations we characterize those g's for which μ(K\cap g(K) > 0, and we also get results about those $g$'s for which $g(K)\su K$ or $g(K)\supset K$ holds.

math.GM↗