Searcharxiv⌕ Search

arXiv subjects

Pavol Zlatoš

Publications and source records attributed to Pavol Zlatoš.

5 recordsLinked to original sources
arXiv2025

Schur ultrafilters and Bohr compactifications of topological groups

In this paper we investigate Schur ultrafilters on groups. Using the algebraic structure of Stone-Čech compactifications of discrete groups and Schur ultrafilters, we give a new description of Bohr compactifications of topological groups. This approach allows us to characterize chart groups that are topological groups. Namely, a chart group $G$ is a topological group if and only if each Schur ultrafilter on $G$ converges to the unit of $G$.

math.GN↗
arXiv2018

A uniform stability principle for dual lattices

We prove a highly uniform stability or "almost-near" theorem for dual lattices of lattices $L \subseteq \Bbb R^n$. More precisely, we show that, for a vector $x$ from the linear span of a lattice $L \subseteq \Bbb R^n$, subject to $λ_1(L) \ge λ> 0$, to be $\varepsilon$-close to some vector from the dual lattice $L'$ of $L$, it is enough that the inner products $u\,x$ are $δ$-close (with $δ< 1/3$) to some integers for all vectors $u \in L$ satisfying $\| u \| \le r$, where $r > 0$ depends on $n$, $λ$, $δ$ and $\varepsilon$, only. This generalizes an earlier analogous result proved for integral vector lattices by M. Mačaj and the second author. The proof is nonconstructive, using the ultraproduct construction and a slight portion of nonstandard analysis.

math.NT↗
arXiv2018

The Bohr compactification of an abelian group as a quotient of its Stone-Čech compactification

We will prove that, for any abelian group $G$, the canonical (surjective and continuous) mapping $\boldsymbolβG \to {\frak b}G$ from the Stone-Čech compactification $\boldsymbolβG$ of $G$ to its Bohr compactfication ${\frak b}G$ is a homomorphism with respect to the semigroup operation on $\boldsymbolβG$, extending the multiplication on $G$, and the group operation on ${\frak b}G$. Moreover, the Bohr compactification ${\frak b}G$ is canonically isomorphic (both in algebraic and topological sense) to the quotient of $\boldsymbolβG$ with respect to the least closed congruence relation on $\boldsymbolβG$ merging all the Schur ultrafilters on $G$ into the unit of $G$.

math.GN↗
arXiv2015

Stability of systems of general functional equations in the compact-open topology

We introduce a fairly general concept of functional equation for $k$-tuples of functions $f_1,\dots,f_k\colon X \to Y$ between arbitrary sets. The homomorphy equations for mappings between groups and other algebraic systems, as well as various types of functional equations and recursion formulas occurring in mathematical analysis or combinatorics, respectively, become special cases (of systems) of such equations. Assuming that $X$ is a locally compact and $Y$ is a completely regular topological space, we show that systems of such functional equations, with parameters satisfying rather a modest continuity condition, are stable in the following intuitive sense: Every $k$-tuple of ``sufficiently continuous,'' ``reasonably bounded'' functions $X \to Y$ satisfying the given system with a ``sufficient precision'' on a ``big enough'' compact set is already ``arbitrarily close'' on an ``arbitrarily big'' compact set to a $k$-tuple of continuous functions solving the system. The result is derived as a consequence of certain intuitively appealing ``almost-near'' principle using the relation of infinitesimal nearness formulated in terms of nonstandard analysis.

math.FA↗