arXiv subjects

Petra Staynova

Publications and source records attributed to Petra Staynova.

6 recordsLinked to original sources

arXiv2023

Monochromatic arithmetic progressions in automatic sequences with group structure

We determine asymptotic growth rates for lengths of monochromatic arithmetic progressions in certain automatic sequences. In particular, we look at (one-sided) fixed points of aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue--Morse and Rudin--Shapiro substitutions, respectively. For such infinite words, we show that there exists a subsequence $\left\{d_n\right\}$ of differences along which the maximum length $A(d_n)$ of a monochromatic arithmetic progression (with fixed difference $d_n$) grows at least polynomially in $d_n$. Explicit upper and lower bounds for the growth exponent can be derived from a finite group associated to the substitution. As an application, we obtain bounds for a van der Waerden-type number for a class of colourings parametrised by the size of the alphabet and the length of the substitution.

math.CO↗

arXiv2021

On long arithmetic progressions in binary Morse-like words

We present results on the existence of long arithmetic progressions in the Thue-Morse word and in a class of generalised Thue-Morse words. Our arguments are inspired by van der Waerden's proof for the existence of arbitrary long monochromatic arithmetic progressions in any finite colouring of the (positive) integers.

math.CO↗

arXiv2019

The Ellis Semigroup of a Generalised Morse System

In this article, we calculate the Ellis semigroup of a certain class of constant length substitutions. This generalises a result of Haddad and Johnson [HJ97] from the binary case to substitutions over arbitrarily large finite alphabets. Moreover, we provide a class of counter-examples to one of the propositions in their paper, which is central to the proof of their main theorem. We give an alternative approach to their result, which centers on the properties of the Ellis semigroup. To do this, we also show a new way to construct an AI tower to the maximal equicontinuous factor of these systems, which gives a more particular approach than the one given by Dekking [Dek78].

math.DS↗

arXiv2012

A Note on the Hausdorff Number of Compact Topological Spaces

The notion of Hausdorff number of a topological space is first introduced in \cite{bonan}, with the main objective of using this notion to obtain generalizations of some known bounds for cardinality of topological spaces. Here we consider this notion from a topological point of view and examine interrelations of the Hausdorff number with compactness.

math.GN↗

arXiv2012

A Comparison of Lindelof-Type Covering Properties of Topological Spaces

Lindelöf spaces are studied in any basic Topology course. However, there are other interesting covering properties with similar behaviour, such as almost Lindelöf, weakly Lindelöf, and quasi-Lindelöf, that have been considered in various research papers. Here we present a comparison between the standard results on Lindelöf spaces and analogous results for weakly and almost Lindelöf spaces. Some theorems, similar to the published ones, will be proved. We also consider counterexamples, most of which have not been included in the standard Topological textbooks, that show the interrelations between those properties and various basic topological notions, such as separability, separation axioms, first countability, and others. Some new features of those examples will be noted in view of the present comparison. We also pose several open questions.

math.GN↗

arXiv2012

A Note on Quasi-Lindelof Spaces

The quasi-Lindelöf property was first introduced by Arhangelski in \cite{Arc}, as a strengthening of the weakly Lindelöf property. However, unlike Lindelöf and weakly Lindelöf spaces, very little is known about how quasi-Lindelöf spaces behave under the main topological operations, and how the property relates to separation axioms. In the present paper, we look at several properties of quasi-Lindelöf spaces. We consider several examples: a weakly Lindelöf space which is not quasi-Lindelöf, a product of Lindelöf spaces which is not even quasi-Lindelöf, and a quasi-Lindelöf space which is not ccc. At the end, we pose some open questions.

math.GN↗