arXiv subjects

Paolo Leonetti

Publications and source records attributed to Paolo Leonetti.

At least 73 records · Page 4Linked to original sources

arXiv2021

A characterization of $(\mathcal{I}, \mathcal{J})$-regular matrices

Let $\mathcal{I},\mathcal{J}$ be two ideals on $\mathbf{N}$ which contain the family $\mathrm{Fin}$ of finite sets. We provide necessary and sufficient conditions on the entries of an infinite real matrix $A=(a_{n,k})$ which maps $\mathcal{I}$-convergent bounded sequences into $\mathcal{J}$-convergent bounded sequences and preserves the corresponding ideal limits. The well-known characterization of regular matrices due to Silverman--Toeplitz corresponds to the case $\mathcal{I}=\mathcal{J}=\mathrm{Fin}$. Lastly, we provide some applications to permutation and diagonal matrices, which extend several known results in the literature.

math.FA↗

arXiv2021

Turnpike in infinite dimension

Let $Φ$ be a correspondence from a normed vector space $X$ into itself, let $u: X\to \mathbf{R}$ be a function, and $\mathcal{I}$ be an ideal on $\mathbf{N}$. Also, assume that the restriction of $u$ on the fixed points of $Φ$ has a unique maximizer $η^\star$. Then, we consider feasible paths $(x_0,x_1,\ldots)$ with values in $X$ such that $x_{n+1} \in Φ(x_n)$ for all $n\ge 0$. Under certain additional conditions, we prove the following turnpike result: every feasible path $(x_0,x_1,\ldots)$ which maximizes the smallest $\mathcal{I}$-cluster point of the sequence $(u(x_0),u(x_1),\ldots)$ is necessarily $\mathcal{I}$-convergent to $η^\star$. We provide examples that, on the one hand, justify the hypotheses of our result and, on the other hand, prove that we are including new cases which were previously not considered in the related literature.

math.FA↗

arXiv2020

A Characterization of Cesaro Convergence

We show that a real bounded sequence $(x_n)$ is Cesàro convergent to $\ell$ if and only if the sequence of averages with indices in $[α^k,α^{k+1})$ converges to $\ell$ for all $α>1$. If, in addition, the sequence $(x_n)$ is nonnegative, then it is Cesàro convergent to $0$ if and only if the same condition holds for some $α>1$.

math.CA↗

arXiv2020

Tauberian theorems for ordinary convergence

We show that a real sequence $x$ is convergent if and only if there exist a regular matrix $A$ and an $F_{σδ}$-ideal $\mathcal{I}$ on $\mathbf{N}$ such that the set of subsequences $y$ of $x$ for which $Ay$ is $\mathcal{I}$-convergent is of the second Baire category. This includes the cases where $\mathcal{I}$ is the ideal of asymptotic density zero sets, the ideal of Banach density zero sets, and the ideal of finite sets. The latter recovers an old result given by Keogh and Petersen in [J. London Math. Soc. \textbf{33} (1958), 121--123]. Our proofs are of a different nature and rely on recent results in the context of $\mathcal{I}$-Baire classes and filter games. As application, we obtain a stronger version of the classical Steinhaus' theorem: for each regular matrix $A$, there exists a $\{0,1\}$-valued sequence $x$ such that $Ax$ is not statistically convergent.

math.FA↗

arXiv2020

Convergent subseries of divergent series

Let $\mathscr{X}$ be the set of positive real sequences $x=(x_n)$ such that the series $\sum_n x_n$ is divergent. For each $x \in \mathscr{X}$, let $\mathcal{I}_x$ be the collection of all $A\subseteq \mathbf{N}$ such that the subseries $\sum_{n \in A}x_n$ is convergent. Moreover, let $\mathscr{A}$ be the set of sequences $x \in \mathscr{X}$ such that $\lim_n x_n=0$ and $\mathcal{I}_x\neq \mathcal{I}_y$ for all sequences $y=(y_n) \in \mathscr{X}$ with $\liminf_n y_{n+1}/y_n>0$. We show that $\mathscr{A}$ is comeager and that contains uncountably many sequences $x$ which generate pairwise nonisomorphic ideals $\mathcal{I}_x$. This answers, in particular, an open question recently posed by M. Filipczak and G. Horbaczewska.

math.CA↗

arXiv2020

On Small Sets of Integers

An upper quasi-density on $\bf H$ (the integers or the non-negative integers) is a real-valued subadditive function $μ^\ast$ defined on the whole power set of $\mathbf H$ such that $μ^\ast(X) \le μ^\ast({\bf H}) = 1$ and $μ^\ast(k \cdot X + h) = \frac{1}{k}\, μ^\ast(X)$ for all $X \subseteq \bf H$, $k \in {\bf N}^+$, and $h \in \bf N$, where $k \cdot X := \{kx: x \in X\}$; and an upper density on $\bf H$ is an upper quasi-density on $\bf H$ that is non-decreasing with respect to inclusion. We say that a set $X \subseteq \bf H$ is small if $μ^\ast(X) = 0$ for every upper quasi-density $μ^\ast$ on $\bf H$. Main examples of upper densities are given by the upper analytic, upper Banach, upper Buck, and upper Pólya densities, along with the uncountable family of upper $α$-densities, where $α$ is a real parameter $\ge -1$ (most notably, $α= -1$ corresponds to the upper logarithmic density, and $α= 0$ to the upper asymptotic density). It turns out that a subset of $\bf H$ is small if and only if it belongs to the zero set of the upper Buck density on $\bf Z$. This allows us to show that many interesting sets are small, including the integers with less than a fixed number of prime factors, counted with multiplicity; the numbers represented by a binary quadratic form with integer coefficients whose discriminant is not a perfect square; and the image of $\bf Z$ through a non-linear integral polynomial in one variable.

math.NT↗

arXiv2020

The Baire Category of Subsequences and Permutations which preserve Limit Points

Let $\mathcal{I}$ be a meager ideal on $\mathbf{N}$. We show that if $x$ is a sequence with values in a separable metric space then the set of subsequences [resp. permutations] of $x$ which preserve the set of $\mathcal{I}$-cluster points of $x$ is topologically large if and only if every ordinary limit point of $x$ is also an $\mathcal{I}$-cluster point of $x$. The analogue statement fails for all maximal ideals. This extends the main results in [Topology Appl. \textbf{263} (2019), 221--229]. As an application, if $x$ is a sequence with values in a first countable compact space which is $\mathcal{I}$-convergent to $\ell$, then the set of subsequences [resp. permutations] which are $\mathcal{I}$-convergent to $\ell$ is topologically large if and only if $x$ is convergent to $\ell$ in the ordinary sense. Analogous results hold for $\mathcal{I}$-limit points, provided $\mathcal{I}$ is an analytic P-ideal.

math.GN↗

arXiv2019

Behavioral Equivalence of Extensive Game Structures

Two extensive game structures with imperfect information are said to be behaviorally equivalent if they share the same map (up to relabelings) from profiles of structurally reduced strategies to induced terminal paths. We show that this is the case if and only if one can be transformed into the other through a composition of two elementary transformations, commonly known as \textquotedblleft Interchanging of Simultaneous Moves\textquotedblright\ and \textquotedblleft Coalescing Moves/Sequential Agent Splitting.\textquotedblright

econ.TH↗

arXiv2019

On the number of gaps of sequences with Poissonian Pair Correlations

A sequence $(x_n)$ on the torus is said to have Poissonian pair correlations if $\# \{1\le i\neq j\le N: |x_i-x_j| \le s/N\}=2sN(1+o(1))$ for all reals $s>0$, as $N\to \infty$. It is known that, if $(x_n)$ has Poissonian pair correlations, then the number $g(n)$ of different gap lengths between neighboring elements of $\{x_1,\ldots,x_n\}$ cannot be bounded along every index subsequence $(n_t)$. First, we improve this by showing that the maximum among the multiplicities of the neighboring gap lengths of $\{x_1,\ldots,x_n\}$ is $o(n)$, as $n\to \infty$. Furthermore, we show that, for every function $f: \mathbf{N}^+\to \mathbf{N}^+$ with $\lim_n f(n)=\infty$, there exists a sequence $(x_n)$ with Poissonian pair correlations and such that $g(n) \le f(n)$ for all sufficiently large $n$. This answers negatively a question posed by G. Larcher.

math.NT↗

arXiv2019

A Tauberian theorem for ideal statistical convergence

Given an ideal $\mathcal{I}$ on the positive integers, a real sequence $(x_n)$ is said to be $\mathcal{I}$-statistically convergent to $\ell$ provided that $$ \textstyle \left\{n \in \mathbf{N}: \frac{1}{n}|\{k \le n: x_k \notin U\}| \ge \varepsilon\right\} \in \mathcal{I} $$ for all neighborhoods $U$ of $\ell$ and all $\varepsilon>0$. First, we show that $\mathcal{I}$-statistical convergence coincides with $\mathcal{J}$-convergence, for some unique ideal $\mathcal{J}=\mathcal{J}(\mathcal{I})$. In addition, $\mathcal{J}$ is Borel [analytic, coanalytic, respectively] whenever $\mathcal{I}$ is Borel [analytic, coanalytic, resp.]. Then we prove, among others, that if $\mathcal{I}$ is the summable ideal $\{A\subseteq \mathbf{N}: \sum_{a \in A}1/a<\infty\}$ or the density zero ideal $\{A\subseteq \mathbf{N}: \lim_{n\to \infty} \frac{1}{n}|A\cap [1,n]|=0\}$ then $\mathcal{I}$-statistical convergence coincides with statistical convergence. This can be seen as a Tauberian theorem which extends a classical theorem of Fridy. Lastly, we show that this is never the case if $\mathcal{I}$ is maximal.

math.FA↗

arXiv2019

On the notions of upper and lower density

Let $\mathcal{P}({\bf N})$ be the power set of ${\bf N}$. We say that a function $μ^\ast: \mathcal{P}({\bf N}) \to \bf R$ is an upper density if, for all $X,Y\subseteq{\bf N}$ and $h, k\in{\bf N}^+$, the following hold: (F1) $μ^\ast({\bf N}) = 1$; (F2) $μ^\ast(X) \le μ^\ast(Y)$ if $X \subseteq Y$; (F3) $μ^\ast(X \cup Y) \le μ^\ast(X) + μ^\ast(Y)$; (F4) $μ^\ast(k\cdot X) = \frac{1}{k} μ^\ast(X)$, where $k \cdot X:=\{kx: x \in X\}$; (F5) $μ^\ast(X + h) = μ^\ast(X)$. We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Polya, and upper analytic densities, together with all upper $α$-densities (with $α$ a real parameter $\ge -1$), are upper densities in the sense of our definition. Moreover, we establish the mutual independence of axioms (F1)-(F5), and we investigate various properties of upper densities (and related functions) under the assumption that (F2) is replaced by the weaker condition that $μ^\ast(X)\le 1$ for every $X\subseteq{\bf N}$. Overall, this allows us to extend and generalize results so far independently derived for some of the classical upper densities mentioned above, thus introducing a certain amount of unification into the theory.

math.NT↗

arXiv2019

Directions sets: A generalization of ratio sets

For every integer $k \geq 2$ and every $A \subseteq \mathbb{N}$, we define the \emph{$k$-directions sets} of $A$ as $D^k(A) := \{{\bf a} / \|{\bf a}\| : {\bf a} \in A^k\}$ and $D^{\underline{k}}(A) := \{{\bf a} / \|{\bf a}\| : {\bf a} \in A^{\underline{k}}\}$, where $\|\cdot\|$ is the Euclidean norm and $A^{\underline{k}} := \{{\bf a} \in A^k : a_i \neq a_j \text{ for all } i \neq j\}$. Via an appropriate homeomorphism, $D^k(A)$ is a generalization of the \emph{ratio set} $R(A) := \{a / b : a,b \in A\}$, which has been studied by many authors. We study $D^k(A)$ and $D^{\underline{k}}(A)$ as subspaces of $S^{k-1} := \{{\bf x} \in [0,1]^k : \|{\bf x}\| = 1\}$. In~particular, generalizing a result of Bukor and Tóth, we provide a characterization of the sets $X \subseteq S^{k-1}$ such that there exists $A \subseteq \mathbb{N}$ satisfying $D^{\underline{k}}(A)^\prime = X$, where $Y^\prime$ denotes the set of accumulation points of $Y$. Moreover, we provide a simple sufficient condition for $D^k(A)$ to be dense in $S^{k-1}$. We conclude leaving some questions for further research.

math.NT↗

arXiv2019

Limit points of subsequences

Let $x$ be a sequence taking values in a separable metric space and $\mathcal{I}$ be a generalized density ideal or an $F_σ$-ideal on the positive integers (in particular, $\mathcal{I}$ can be any Erd{\H o}s--Ulam ideal or any summable ideal). It is shown that the collection of subsequences of $x$ which preserve the set of $\mathcal{I}$-cluster points of $x$ [respectively, $\mathcal{I}$-limit points] is of second category if and only if the set of $\mathcal{I}$-cluster points of $x$ [resp., $\mathcal{I}$-limit points] coincides with the set of ordinary limit points of $x$; moreover, in this case, it is comeager. In particular, it follows that the collection of subsequences of $x$ which preserve the set of ordinary limit points of $x$ is comeager.

math.GN↗

arXiv2019

The general linear equation on open connected sets

Fix non-zero reals $α_1,\ldots,α_n$ with $n\ge 2$ and let $K$ be a non-empty open connected set in a topological vector space such that $\sum_{i\le n}α_iK\subseteq K$ (which holds, in particular, if $K$ is an open convex cone and $α_1,\ldots,α_n>0$). Let also $Y$ be a vector space over $\mathbb{F}:=\mathbb{Q}(α_1,\ldots,α_n)$. We show, among others, that a function $f: K\to Y$ satisfies the general linear equation $$ \textstyle \forall x_1,\ldots,x_n \in K,\,\,\,\,\, f\left(\sum_{i\le n}α_i x_i\right)=\sum_{i\le n}α_i f(x_i) $$ if and only if there exist a unique $\mathbb{F}$-linear $A:X\to Y$ and unique $b\in Y$ such that $f(x)=A(x)+b$ for all $x \in K$, with $b=0$ if $\sum_{i\le n}α_i\neq 1$. The main tool of the proof is a general version of a result Radó and Baker on the existence and uniqueness of extension of the solution on the classical Pexider equation.

math.FA↗

arXiv2019

Practical numbers among the binomial coefficients

A "practical number" is a positive integer $n$ such that every positive integer less than $n$ can be written as a sum of distinct divisors of $n$. We prove that most of the binomial coefficients are practical numbers. Precisely, letting $f(n)$ denote the number of binomial coefficients $\binom{n}{k}$, with $0 \leq k \leq n$, that are not practical numbers, we show that \begin{equation*} f(n) < n^{1 - (\log 2 - δ)/\log \log n} \end{equation*} for all integers $n \in [3, x]$, but at most $O_γ(x^{1 - (δ- γ) / \log \log x})$ exceptions, for all $x \geq 3$ and $0 < γ< δ< \log 2$. Furthermore, we prove that the central binomial coefficient $\binom{2n}{n}$ is a practical number for all positive integers $n \leq x$ but at most $O(x^{0.88097})$ exceptions. We also pose some questions on this topic.

math.NT↗

arXiv2019

Characterizations of the Ideal Core

Given an ideal $\mathcal{I}$ on $ω$ and a sequence $x$ in a topological vector space, we let the $\mathcal{I}$-core of $x$ be the least closed convex set containing $\{x_n: n \notin I\}$ for all $I \in \mathcal{I}$. We show two characterizations of the $\mathcal{I}$-core. This implies that the $\mathcal{I}$-core of a bounded sequence in $\mathbf{R}^k$ is simply the convex hull of its $\mathcal{I}$-cluster points. As applications, we simplify and extend several results in the context of Pringsheim-convergence and $e$-convergence of double sequences.

math.FA↗

arXiv2019

Characterizations of Ideal Cluster Points

Given an ideal $\mathcal{I}$ on $ω$, we prove that a sequence in a topological space $X$ is $\mathcal{I}$-convergent if and only if there exists a ``big'' $\mathcal{I}$-convergent subsequence. Then, we study several properties and show two characterizations of the set of $\mathcal{I}$-cluster points as classical cluster points of a filters on $X$ and as the smallest closed set containing ``almost all'' the sequence. As a consequence, we obtain that the underlying topology $τ$ coincides with the topology generated by the pair $(τ,\mathcal{I})$.

math.CA↗

arXiv2018

Invariance of Ideal Limit Points

Let $\mathcal{I}$ be an analytic P-ideal [respectively, a summable ideal] on the positive integers and let $(x_n)$ be a sequence taking values in a metric space $X$. First, it is shown that the set of ideal limit points of $(x_n)$ is an $F_σ$-set [resp., a closet set]. Let us assume that $X$ is also separable and the ideal $\mathcal{I}$ satisfies certain additional assumptions, which however includes several well-known examples, e.g., the collection of sets with zero asymptotic density, sets with zero logarithmic density, and some summable ideals. Then, it is shown that the set of ideal limit points of $(x_n)$ is equal to the set of ideal limit points of almost all its subsequences.

math.CA↗