Searcharxiv⌕ Search

arXiv subjects

Szymon Glab

Publications and source records attributed to Szymon Glab.

11 recordsLinked to original sources
arXiv2025

On the complexity of upper frequently hypercyclic vectors

Given a continuous linear operator $T:X\to X$, where $X$ is a topological vector space, let $\mathrm{UFHC}(T)$ be the set of upper frequently hypercyclic vectors, that is, the set of vectors $x \in X$ such that $\{n \in ω: T^nx \in U\}$ has positive upper asymptotic density for all nonempty open sets $U\subseteq X$. It is known that $\mathrm{UFHC}(T)$ is a $G_{δσδ}$-set which is either empty or contains a dense $G_δ$-set. Using a purely topological proof, we improve it by showing that $\mathrm{UFHC}(T)$ is always a $G_{δσ}$-set. Bonilla and Grosse-Erdmann asked in [Rev. Mat. Complut. \textbf{31} (2018), 673--711] whether $\mathrm{UFHC}(T)$ is always a $G_δ$-set. We answer such question in the negative, by showing that there exists a continuous linear operator $T$ for which $\mathrm{UFHC}(T)$ is not a $F_{σδ}$-set (hence not $G_δ$). In addition, we study the [non-]equivalence between (the ideal versions of) upper frequently hypercyclicity in the product topology and upper frequently hypercyclicity in the norm topology.

math.FA↗
arXiv2024

Topological complexity of ideal limit points

Given an ideal $\mathcal{I}$ on the nonnegative integers $ω$ and a Polish space $X$, let $\mathscr{L}(\mathcal{I})$ be the family of subsets $S\subseteq X$ such that $S$ is the set of $\mathcal{I}$-limit points of some sequence taking values in $X$. First, we show that $\mathscr{L}(\mathcal{I})$ may attain arbitrarily large Borel complexity. Second, we prove that if $\mathcal{I}$ is a $G_{δσ}$-ideal then all elements of $\mathscr{L}(\mathcal{I})$ are closed. Third, we show that if $\mathcal{I}$ is a simply coanalytic ideal and $X$ is first countable, then every element of $\mathscr{L}(\mathcal{I})$ is simply analytic. Lastly, we studied certain structural properties and the topological complexity of minimal ideals $\mathcal{I}$ for which $\mathscr{L}(\mathcal{I})$ contains a given set.

math.GN↗
arXiv2022

Set of uniqueness for Cantorvals

The main result is that the celebrated Guthrie-Nymann's Cantorval has comeager set of uniqueness. On the other hand many other Cantorvals have meager set of uniqueness.

math.CA↗
arXiv2021

Another characterization of meager ideals

We show that an ideal $\mathcal{I}$ on the positive integers is meager if and only if there exists a bounded nonconvergent real sequence $x$ such that the set of subsequences [resp. permutations] of $x$ which preserve the set of $\mathcal{I}$-limit points is comeager and, in addition, every accumulation point of $x$ is also an $\mathcal{I}$-limit point (that is, a limit of a subsequence $(x_{n_k})$ such that $\{n_1,n_2,\ldots,\} \notin \mathcal{I}$). The analogous characterization holds also for $\mathcal{I}$-cluster points.

math.GN↗
arXiv2017

Levy-Steinitz theorem and achievement sets of conditionally convergent series on the real plane

Levy-Steinitz theorem characterize sum range of conditionally convergent series, that is a set of all its convergent rearrangements; in finitely dimensional spaces -- it is an affine subspace. An achievement of a series is a set of all its subsums. We study the properties of achievement sets of series whose sum range is the whole plane. It turns out that it varies on the number of Levy vectors of a series.

math.FA↗
arXiv2012

Positive solutions for anisotropic discrete BVP

Using mountain pass arguments and the Karsuh-Kuhn-Tucker Theorem, we prove the existence of at least two positive solution of the anisotropic discrete Dirichlet boundary value problem. Our results generalize and improve those of [15].

math.CA↗
arXiv2012

Large structures made of nowhere $L^p$ functions

We say that a real-valued function $f$ defined on a positive Borel measure space $(X,μ)$ is nowhere $q$-integrable if, for each nonvoid open subset $U$ of $X$, the restriction $f|_U$ is not in $L^q(U)$. When $(X,μ)$ satisfies some natural properties, we show that certain sets of functions defined in $X$ which are $p$-integrable for some $p$'s but nowhere $q$-integrable for some other $q$'s ($0<p,q<\infty$) admit a variety of large linear and algebraic structures within them. The presented results answer a question from Bernal-González, improve and complement recent spaceability and algebrability results from several authors and motivates new research directions in the field of spaceability.

math.FA↗
arXiv2012

The continuity properties of compact-preserving functions

A function $f:X\to Y$ between topological spaces is called {\em compact-preserving} if the image $f(K)$ of each compact subset $K\subset X$ is compact. We prove that a function $f:X\to Y$ defined on a strong Frechet space $X$ is compact-preserving if and only if for each point $x\in X$ there is a compact subset $K_x\subset Y$ such that for each neighborhood $O_{f(x)}\subset Y$ of $f(x)$ there is a neighborhood $O_x\subset X$ of $x$ such that $f(O_x)\subset O_{f(x)}\cup K_x$ and the set $K_x\setminus O_{f(x)}$ is finite. This characterization is applied to give an alternative proof of a classical characterization of continuous functions on locally connected metrizable spaces as functions that preserve compact and connected sets. Also we show that for each compact-preserving function $f:X\to Y$ defined on a (strong) Fréchet space $X$, the restriction $f|LI'_f$ (resp. $f|LI_f)$ is continuous. Here $LI_f$ is the set of points $x\in X$ of local infinity of $f$ and $LI'_f$ is the set of non-isolated points of the set $LI_f$. Suitable examples show that the obtained results cannot be improved.

math.GN↗
arXiv2012

Spaceability and algebrability of sets of nowhere integrable functions

We show that the set of Lebesgue integrable functions in $[0,1]$ which are nowhere essentially bounded is spaceable, improving a result from [F. J. García-Pacheco, M. Martín, and J. B. Seoane-Sepúlveda. \textit{Lineability, spaceability, and algebrability of certain subsets of function spaces,} Taiwanese J. Math., \textbf{13} (2009), no. 4, 1257--1269], and that it is strongly $\mathfrak{c}$-algebrable. We prove strong $\mathfrak{c}$-algebrability and non-separable spaceability of the set of functions of bounded variation which have a dense set of jump discontinuities. Applications to sets of Lebesgue-nowhere-Riemann integrable and Riemann-nowhere-Newton integrable functions are presented as corollaries. In addition we prove that the set of Kurzweil integrable functions which are not Lebesgue integrable is spaceable (in the Alexievicz norm) but not 1-algebrable. We also show that there exists an infinite dimensional vector space $S$ of differentiable functions such that each element of the $C([0,1])$-closure of $S$ is a primitive to a Kurzweil integrable function, in connection to a classic spaceability result from [V. I. Gurariy, \textit{Subspaces and bases in spaces of continuous functions (Russian),} Dokl. Akad. Nauk SSSR, \textbf{167} (1966), 971-973].

math.FA↗
arXiv2011

Algebrability, non-linear properties, and special functions

We construct uncountably generated algebras inside the following sets of special functions: Sierpiński-Zygmund functions, perfectly everywhere surjective functions and nowhere continuous Darboux functions. All conclusions obtained in this paper are improvements of some already known results.

math.FA↗