arXiv subjects

Donát Nagy

Publications and source records attributed to Donát Nagy.

6 recordsLinked to original sources

arXiv2023

Compact sets with large projections and nowhere dense sumset

We answer a question of Banakh, Jabłońska and Jabłoński by showing that for $d\ge 2$ there exists a compact set $K \subseteq \mathbb{R}^d$ such that the projection of $K$ onto each hyperplane is of non-empty interior, but $K+K$ is nowhere dense. The proof relies on a random construction. A natural approach in the proofs is to construct such a $K$ in the unit cube with full projections, that is, such that the projections of $K$ agree with that of the unit cube. We investigate the generalization of these problems for projections onto various dimensional subspaces as well as for $\ell$-fold sumsets. We obtain numerous positive and negative results, but also leave open many interesting cases. We also show that in most cases if we have a specific example of such a compact set then actually the generic (in the sense of Baire category) compact set in a suitably chosen space is also an example. Finally, utilizing a computer-aided construction, we show that the compact set in the plane with full projections and nowhere dense sumset can be self-similar.

math.CA↗

arXiv2020

Games characterizing limsup functions and Baire class 1 functions

We consider a real-valued function $f$ defined on the set of infinite branches $X$ of a countably branching pruned tree $T$. The function $f$ is said to be a \textit{limsup function} if there is a function $u \colon T \to \mathbb{R}$ such that $f(x) = \limsup_{t \to \infty} u(x_{0},\dots,x_{t})$ for each $x \in X$. We study a game characterization of limsup functions, as well as a novel game characterization of functions of Baire class 1.

math.GN↗

arXiv2020

Substituting the typical compact sets into a power series

The Minkowski sum and Minkowski product can be considered as the addition and multiplication of subsets of $\mathbb{R}$. If we consider a compact subset $K\subseteq[0,1]$ and a power series $f$ which is absolutely convergent on $[0,1]$, then we may use these operations and the natural topology of the space of compact sets to substitute the compact set $K$ into the power series $f$. Changhao Chen studied this kind of substitution in the special case of polynomials and showed that if we substitute the typical compact set $K\subseteq [0,1]$ into a polynomial, we get a set of Hausdorff dimension 0. We generalize this result and show that the situation is the same for power series where the coefficients converge to zero quickly. On the other hand we also show a large class of power series where the result of the substitution has Hausdorff dimension one.

math.CA↗

arXiv2019

A Haar meager set that is not strongly Haar meager

Following Darji, we say that a Borel subset $B$ of an abelian Polish group $G$ is Haar meager if there is a compact metric space $K$ and a continuous function $f : K \to G$ such that the preimage of the translate, $f^{-1}(B+g)$ is meager in $K$ for every $g \in G$. The set $B$ is called strongly Haar meager if there is a compact set $C \subseteq G$ such that $(B+g) \cap C$ is meager in $C$ for every $g \in G$. The main open problem in this area is Darji's question asking whether these two notions are the same. Even though there have been several partial results suggesting a positive answer, in this paper we construct a counterexample. More specifically, we construct a $G_δ$ set in $\mathbb{Z}^ω$ that is Haar meager but not strongly Haar meager. We also show that no $F_σ$ counterexample exists, hence our result is optimal.

math.LO↗

arXiv2018

Haar null and Haar meager sets: a survey and new results

We survey results about Haar null subsets of (not necessarily locally compact) Polish groups. The aim of this paper is to collect the fundamental properties of the various possible definitions of Haar null sets, and also to review the techniques that may enable the reader to prove results in this area. We also present several recently introduced ideas, including the notion of Haar meager sets, which are closely analogous to Haar null sets. We prove some results in a more general setting than that of the papers where they were originally proved and prove some results for Haar meager sets which were already known for Haar null sets.

math.LO↗

arXiv2018

Low complexity Haar null sets without G_δ hulls in Z^ω

We show that for every $2\le ξ<ω_1$ there exists a Haar null set in $\mathbb{Z}^ω$ that is the difference of two $\mathbfΠ^0_ξ$ sets but not contained in any $\mathbfΠ^0_ξ$ Haar null set. In particular, there exists a Haar null set in $\mathbb{Z}^ω$ that is the difference of two $G_δ$ sets but not contained in any $G_δ$ Haar null set. This partially answers a question of M. Elekes and Z. Vidnyánszky. To prove this, we also prove a theorem which characterizes the Haar null subsets of $\mathbb{Z}^ω$.

math.LO↗