Články, ktoré by sa hodili na seminár

[Martin Sleziak]

Tu by sme mohli skúsiť pozbierať nejaké návrhy na články, ktoré by mohli byť zaujímavé v súvislosti s témami, ktorými sa venujeme na seminári.

Časom možno niektoré z nich vyberieme a skúsime na seminári zreferovať.

Na obsiahlejšie témy (t.j. veci, kde by nešlo priamo iba o študovanie jedného konkrétneho článku) som otvoril samostatný topic: viewtopic.php?f=41&t=924

[Martin Sleziak]

Paolo Leonetti, Salvatore Tringali: On the notions of upper and lower density, http://arxiv.org/abs/1506.04664

V tomto článku sa autori zaoberajú axiomatickému prístupu k hornej hustote. Definujú niekoľko axióm pre horné hustoty a potom sa pozerajú na to, ktoré vlastnosti hustôt už z týchto axióm vyplývajú.

Je to pomerne rozsiahly článok, čiže ak sa ho rozhodneme zreferovať na seminári, bude treba z neho povyberať vhodné veci.

Tí istý autori majú aj iný článok s príbuznou témou: Upper and lower densities have the strong Darboux property, http://arxiv.org/abs/1510.07473

[Martin Sleziak]

L. Mišík, J. T. Tóth: Large families of almost disjoint large subsets of $\mathbb N$
Acta Univ. Sapientiae, Mathematica, 3, 1 (2011) 26–33
https://www.emis.de/journals/AUSM/math31.htm
https://www.emis.de/journals/AUSM/C3-1/math31-2.pdf

So skoro disjunktnými systémami na $\mathbb N$ sme sa už stretli: viewtopic.php?t=788
V tomto článku sa študuje otázka, či sa dajú zostrojiť skoro disjunktné systémy obsahujúce $\mathfrak c$ množín, pričom tieto množiny budú v nejakom zmysle veľké.
Konkrétne sa článok zaoberá takými množina, kde podielová množin je hustá v $(0,\infty)$. A tiež takými, ktoré majú hornú (váženú) hustotu rovnú $1$.

[Martin Sleziak]

Alain Faisant, Georges Grekos, Ladislav Mišík: Some generalizations of Olivier's theorem.
Ja mám preprint od Slava Mišíka. Nejaký preprint sa dá nájsť aj tu: http://webperso.univ-st-etienne.fr/~gre ... ierSAG.pdf (Ten má starší dátum, ale zatiaľ som detailne neskúmal, či sú medzi nimi rozdiely). Neviem povedať, či to je momentálne niekde zaslané alebo už prijaté, alebo či to dokonca už vyšlo.
Hlavným obsahom článku je niekoľko rôznych možností, ako sa dá definovať I-monotónnosť postupnosti. A tiež sú tam podokazované nejaké verzie Abel-Pringsheim-Olivierovej vety.

[Martin Sleziak]

Vlado Toma navrhol nejaké veci, ktoré počul na konferencii v Starej Lesnej. Na stránke konferencie som nenašiel zborník abstraktov, iba program. Vlado má zborník abstraktov v papierovej podobe, tu je čo tam píšu.
(Tým, že sú to veci z nedávnej konferencie, je dosť pravdepodobné, že veľa z toho ešte zatiaľ nebolo publikované.)

Szymon Glab: Lebesgue density and statistical convergence
Autor má preprint na svojej stránke: http://im0.p.lodz.pl/~sglab/szymon/publikacje.html http://im0.p.lodz.pl/~sglab/szymon/gestosc.pdf

Spoiler:

Let $A\subset\mathbb R$ be measurable, $\lambda$ stands for Lebesgue measure on $\mathbb R$. We say that $0$ is Lebesgue right density point of $A$ if
$$d^+(A,0) = \lim_{h\to0^+} \frac{\lambda(A\cap[0,h])}h=1.$$
Note that
$$
\frac{\lambda(A\cap[0,\frac1{n+1}])}{\frac1{n+1}}\cdot\frac{n}{n+1} \le \frac{\lambda(A\cap[0,h])}h \le \frac{\lambda(A\cap[0,\frac1n])}{\frac1n}\cdot\frac{n+1}n.
$$
Therefore $d^+(A,0)=1$ if and only if
$$\lim\limits_{n\to\infty} n\lambda\left(A\cap[0,1/n]\right)=1.$$
Let $a_n=\lambda(A\cap[1/(n+1),1/n])$. Then
$$\lambda(A\cap[0,1/m]) = \sum_{k=n}^\infty \lambda(A\cap[1/(k+1),1/k])=\sum_{k=n}^\infty a_k.$$
The intuition is the following. If $d^+(A,0)=1$, then $\lambda(A\cap[1/(n+1),1/n])$ should be close to the length $1/(n(n+1))$ of $[1/(n+1),1/n]$ if $n$ tends to $\infty$, and thus $a_n/(1/n-1/(n+1))=n(n+1)a_n$ should tend to $1$.

In fact the following holds.
Theorem 1. $d^+(A,0)=1$ if and only if $n(n+1)a_n$ tends statistically to $1$.

Theorem 1 justifies the following definition. Let $\mathcal I$ be an ideal of subsets of $\mathbb N$. We say that $x$ is an $\mathcal I$-right-density point of a measurable set $A$, in symbols $d_+^{\mathcal I}(A,x)=1$, if $n(n+1)\lambda(A\cap[x+\frac1{n+1},x+\frac1n])$ tends to $1$ with respect to $\mathcal I$. Theorem 1 says that $d_+^{{\mathcal I}_d}(A,x)=1$ $\Longleftrightarrow$ $d_+(A,x)$ where $\mathcal I_d$ stands for the density zero ideal. By $\mathrm{Fin}$ we denote the ideal of finite subsets of $\mathbb N$. Note that if $\mathcal I \subset \mathcal J$, then $d_+^{\mathcal I}(A,x)=1$ implies $d_+^{\mathcal J}(A,x)=1$. The following is the strengthening of Lebesgue's one-dimensional density theorem. We mimic the proof of Lebesgue's density theorem presented by Faure in [1].

Theorem 2. Let $A\subseteq\mathbb R$ be measurable. Then $\lambda(\{x\in A \colon d_+^{\mathrm{Fin}}(A,x)\ne1\})=0$.

We will discuss the result of Wilczynski [2] where the other connection between Lebesgue density points and the statistical convergence is given there.

[1] C. A. Faure, A short proof of Lebesgue's density theorem. Amer. Math. Monthly 109 (2002), no. 2, 194-196. JSTOR, Google
[2] W. Wilczynski. Statistical density points. Folia Math. 11 (2014), no. 1, 59-62. http://fm.math.uni.lodz.pl/artykuly/11/wilczynski.pdf

Michal Poplawski, Marek Balcerzak and Artur Wachowicz: Ideal convergent subseries and rearrangements

Spoiler:

Let $\mathcal I$ be an ideal on $\mathbb N$ with the Baire property and $\sum x_n$ be an absolutely convergent series with real terms. We show that the set of $\mathcal I$ -convergent subseries and $\mathcal I$-convergent rearrangements are of the first category in the respective Polish spaces. Further, we use this fact in analysis of series of real-valued functions with the Baire property assuming that $\mathcal I$ is analytic or coanalytic.

Artur Wachowicz: Ideal convergent subsequences and rearrangements for divergent sequences of numbers and functions

Spoiler:

We investigate the Baire category of $\mathcal I$-convergent subsequences and rearrangement of a divergent sequence $s=(s_n)$ of reals if $\mathcal I$ is an ideal on $\mathbb N$ having the Baire property. We also discuss the measure of the set of $\mathcal I$-convergent subsequences for some classes of ideals on $\mathbb N$. Our result generalizes theorems due to H. Miller and C. Orhan (cf. [8], 2001).

Moreover, for a sequence of functions $(f_n)$ (from a Polish space $X$ into a complete metric space $Z$) with the Baire property, which is divergent on a comeager set, we investigate the Baire category of $\mathcal I$-convergent subsequences and rearrangements of $(f_n)$ if $\mathcal I$ is an analytic or coanalytic ideal on $\mathbb N$.

Our results generalize a theorem of Kallman (cf. [5], 1999). A similar theorem for subsequences is obtained if $(X,\mu)$ is a $\sigma$-finite complete measure space and a sequence $(f_n)$ of measurable functions from $X$ to $Z$ is $\mathcal I$-divergent $\mu$-alsmost everywhere. Then the set of subsequences of $(f_n)$, $\mathcal I$-divergent $\mu$-almost everywhere, is of full product measure in $\{0,1\}^{\mathbb N}$ (here we ssume in addition that $\mathcal I$ has the property (G)).

[1] M. Balcerzak, Sz. Glab, J. Swaczyna, Ideal invariant injections, submitted.
[2] K. Dems, On I-Cauchy sequences, Real Anal. Exchange 30 (2004/2005), 123–128.
[3] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244.
[4] J.A. Fridy, On statistical convergence, Analysis 5 (1985), 301–313.
[5] R.R. Kallman, Subsequences and category, Internat. J. Math. 22 (1999), 709–712.
[6] A.S. Kechris, Classical Descriptive Set Theory, Springer, New York 1998.
[7] P. Kostyrko, T. Šalát, W. Wilczynski, I-convergence, Real Anal. Exchange 26 (2000-2001), 669–685.
[8] H.I. Miller, C. Orhan, On almost convergent and statistically convergent subsequences, Acta Math. Hungar. 93 (2001), 135-151.
[9] S. Solecki: Analytic ideals and their applications, Ann. Pure. Appl. Logic 99 (1999), 51-72
[10] S.M. Srivastava, A course of Borel sets, Springer, New York 1998.

Našiel som takýto preprint, kde sú ale uvedení traja autori: https://arxiv.org/abs/1604.08359 Zdá sa, že oba referáty uvedené vyššie by sa mohli (aspoň čiastočne) týkať tohoto článku.

[Martin Sleziak]

Pretože súvisí s vecami, ktoré boli inom článku na seminári (a tiež preto, že možno by sa nejaké veci odtiaľ dali zovšeobecniť), padol návrh prečítať si článok:
J. A. Fridy and H. I. Miller: A Matrix Characterization of Statistical Convergence, Analysis, Volume 11, Issue 1 (Mar 1991), pages 59–66. http://dx.doi.org/10.1524/anly.1991.11.1.59

[Martin Sleziak]

Piotr Miska, János T. Tóth: On interesting subsequences of the sequence of primes
https://arxiv.org/abs/1908.10421

Abstract: Denote by $\mathbb N$ and $\mathbb P$ the set of all positive integers and prime numbers, respectively. Let $\mathbb P=\{p_1<p_2<\dots <p_n<\dots\}$, where $p_n$ is an $n-th$ prime number. For $k\in\mathbb N$ we recursively define subsequences $(p^{(k)}_n)_{n=1}^{+\infty}$ of the sequence $(p_n)_{n=1}^{+\infty}$ in the following way: let $p_n^{(1)}=p_n$ and $p_n^{(k+1)}=p_{p_n^{(k)}}$. In this paper we study and describe some interesting properties of the sets $\mathbb P_k=\{p_1^{(k)}<p_2^{(k)}<\dots<p_n^{(k)}<\dots\}$, $\mathbb P_n^T=\{p_n^{(1)}<p_n^{(2)}<\dots<p_n^{(k)}<\dots\}$ and $\text{Diag}\mathbb P=\{p^{(1)}_1<p^{(2)}_2<\dots <p^{(k)}_k<\dots\}$ and their elements, for $k,n\in\mathbb N$.

[Martin Sleziak]

Pratulananda Das, Sayan Sengupta, Szymon Glab, Marek Bienias: Certain aspects of ideal convergence in topological spaces
https://doi.org/10.1016/j.topol.2019.107005

Abstract: In this note we investigate certain aspects of ideal convergence of functions, in a very general context. We consider the situation, given an arbitrary infinite set $S$, free ideals $\mathscr I$, $\mathscr K$ ($\subset\mathscr I$) and a family $\mathscr F\subset[S]^{>\omega}$, when an $\mathscr I$-convergent function $f$ from $S$ to a topological space $X$ will have a $\mathscr F$-subfunction which is convergent to the same limit. We present certain observations on such topological spaces. Further we also introduce the notion of $\mathscr I$-cluster points of functions and make some observations.

[Martin Sleziak]

G.G. Lorentz: A contribution to the theory of divergent sequences; https://projecteuclid.org/euclid.acta/1485888479
Ten článok obsahuje charakterizáciu ohraničených postupností, pre ktoré sa zhodujú hodnoty všetkých Banachových limít - almost convergent sequence.
Abstract: In this paper we define and examine a new method of summation which assigns a general limit $\operatorname{Lim} x_n$ to cer tain bounded sequences $x=\{x_n\}$. This method is analogous to the mean values which are used in the theory of almost periodic functions, furthermore it is narrowly connected with the limits of S. Banach. The sequences which are summable by this method $F$ we shall call almost convergent. In spite of the fact that our method contains certain classes of matrix methods (for bounded sequences) it is not strong (§3). Its most remarkable property is that most of the commonly used matrix methods contain the method $F$ (§5). In spite of this $F$ is equivalent to none of the matrix methods (§7). In §6 we shall examine a certain class of matrix methods and compare them with the method $F$.

Chao You: Advances in almost convergence; https://projecteuclid.org/euclid.afa/1399900023
Abstract: ‎In this paper‎, ‎we first give the concept of properly distributed‎ ‎sequence‎, ‎and prove that it is almost convergent with F-limit‎ ‎expressed as a formal integral‎. ‎Basing on these‎, ‎we review the work‎ ‎of Feng and Li‎, ‎which is shown to be a special case of our‎ ‎generalized theory‎. ‎Then we generalize Banach limit to Banach limit‎ ‎functional‎, ‎which is the minimum requirement to characterize strong‎ ‎almost convergence for bounded sequences in normed vector space‎. ‎With this machinery‎, ‎we show that Hajdukovic's almost‎ ‎convergence and quasi-almost convergence are both equivalent to our‎ ‎strong almost convergence‎.