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 $F_\sigma$-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.Comment: To appear in Topology Appl. arXiv admin note: substantial text overlap with arXiv:1711.0426

A note on primes in certain residue classes

Given positive integers $a_1,\ldots,a_k$, we prove that the set of primes $p$ such that $p \not\equiv 1 \bmod{a_i}$ for $i=1,\ldots,k$ admits asymptotic density relative to the set of all primes which is at least $\prod_{i=1}^k \left(1-\frac{1}{\varphi(a_i)}\right)$, where $\varphi$ is the Euler's totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer $n$ such that $n \not\equiv 0 \bmod a_i$ for $i=1,\ldots,k$ admits asymptotic density which is at least $\prod_{i=1}^k \left(1-\frac{1}{a_i}\right)$

Upper and lower densities have the strong Darboux property

Let $\mathcal{P}({\bf N})$ be the power set of $\bf N$. An upper density (on $\bf N$) is a non\-decreasing and subadditive function $\mu^\ast: \mathcal{P}({\bf N})\to\bf R$ such that $\mu^\ast({\bf N}) = 1$ and $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$ for all $X \subseteq \bf N$ and $h,k \in {\bf N}^+$, where $k \cdot X + h := \{kx + h: x \in X\}$. The upper asymptotic, upper Banach, upper logarithmic, upper Buck, upper P\'olya, and upper analytic densities are examples of upper densities. We show that every upper density $\mu^\ast$ has the strong Darboux property, and so does the associated lower density, where a function $f: \mathcal P({\bf N}) \to \bf R$ is said to have the strong Darboux property if, whenever $X \subseteq Y \subseteq \bf N$ and $a \in [f(X),f(Y)]$, there is a set $A$ such that $X\subseteq A\subseteq Y$ and $f(A)=a$. In fact, we prove the above under the assumption that the monotonicity of $\mu^\ast$ is relaxed to the weaker condition that $\mu^\ast(X) \le 1$ for every $X \subseteq \bf N$.Comment: 10 pages, no figures. Fixed minor details and streamlined the exposition. To appear in Journal of Number Theor

On the greatest common divisor of nn and the nnth Fibonacci number

Let $\mathcal{A}$ be the set of all integers of the form $\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number. We prove that $\#\left(\mathcal{A} \cap [1, x]\right) \gg x / \log x$ for all $x \geq 2$, and that $\mathcal{A}$ has zero asymptotic density. Our proofs rely on a recent result of Cubre and Rouse which gives, for each positive integer $n$, an explicit formula for the density of primes $p$ such that $n$ divides the rank of appearance of $p$, that is, the smallest positive integer $k$ such that $p$ divides $F_k$

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 $\mu^\ast: \mathcal{P}({\bf N}) \to \bf R$ is an upper density if, for all $X,Y\subseteq{\bf N}$ and $h, k\in{\bf N}^+$, the following hold: (F1) $\mu^\ast({\bf N}) = 1$; (F2) $\mu^\ast(X) \le \mu^\ast(Y)$ if $X \subseteq Y$; (F3) $\mu^\ast(X \cup Y) \le \mu^\ast(X) + \mu^\ast(Y)$; (F4) $\mu^\ast(k\cdot X) = \frac{1}{k} \mu^\ast(X)$, where $k \cdot X:=\{kx: x \in X\}$; (F5) $\mu^\ast(X + h) = \mu^\ast(X)$. We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Polya, and upper analytic densities, together with all upper $\alpha$-densities (with $\alpha$ 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 $\mu^\ast(X)\le 1$ 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.Comment: 26 pp, no figs. Added a 'Note added in proof' at the end of Sect. 7 to answer Question 6. Final version to appear in Proc. Edinb. Math. Soc. (the paper is a prequel of arXiv:1510.07473

Characterizations of the Ideal Core

Given an ideal $\mathcal{I}$ on $\omega$ 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.Comment: 10 pages, to appear in Journal of Mathematical Analysis and Application