arXiv subjects
Publications and source records attributed to Paolo Leonetti.
Let $\mathcal{I}\subseteq\mathcal{P}(ω)$ be a meager ideal. Then there are no continuous projections from $\ell_\infty$ onto the set of bounded sequences which are $\mathcal{I}$-convergent to $0$. In particular, it follows that the set of bounded sequences statistically convergent to $0$ is not isomorphic to $\ell_\infty$.
Let $(x_n)$ be a positive real sequence decreasing to $0$ such that the series $\sum_n x_n$ is divergent and $\liminf_{n} x_{n+1}/x_n>1/2$. We show that there exists a constant $θ\in (0,1)$ such that, for each $\ell>0$, there is a subsequence $(x_{n_k})$ for which $\sum_k x_{n_k}=\ell$ and $x_{n_k}=O(θ^k)$.
We define a class of so-called thinnable ideals $\mathcal{I}$ on the positive integers which includes several well-known examples, e.g., the collection of sets with zero asymptotic density, sets with zero logarithmic density, and several summable ideals. Given a sequence $(x_n)$ taking values in a separable metric space and a thinnable ideal $\mathcal{I}$, it is shown that the set of $\mathcal{I}$-cluster points of $(x_n)$ is equal to the set of $\mathcal{I}$-cluster points of almost all its subsequences, in the sense of Lebesgue measure. Lastly, we obtain a characterization of ideal convergence, which improves the main result in [Trans. Amer. Math. Soc. 347 (1995), 1811--1819].
Let $S$ be the set of subsequences $(x_{n_k})$ of a given real sequence $(x_n)$ which preserve the set of statistical cluster points. It has been recently shown that $S$ is a set of full (Lebesgue) measure. Here, on the other hand, we prove that $S$ is meager if and only if there exists an ordinary limit point of $(x_n)$ which is not a statistical cluster point of $(x_n)$. This provides a non-analogue between measure and category.
Given $k, \ell \in {\bf N}^+$, let $x_{i,j}$ be, for $1 \le i \le k$ and $0 \le j \le \ell$, some fixed integers, and define, for every $n \in {\bf N}^+$, $s_n := \sum_{i=1}^k \prod_{j=0}^\ell x_{i,j}^{n^j}$. We prove that the following are equivalent: (a) There are a real $θ> 1$ and infinitely many indices $n$ for which the number of distinct prime factors of $s_n$ is greater than the super-logarithm of $n$ to base $θ$. (b) There do not exist non-zero integers $a_0,b_0,\ldots,a_\ell,b_\ell $ such that $s_{2n}=\prod_{i=0}^\ell a_i^{(2n)^i}$ and $s_{2n-1}=\prod_{i=0}^\ell b_i^{(2n-1)^i}$ for all $n$. We will give two different proofs of this result, one based on a theorem of Evertse (yielding, for a fixed finite set of primes $\mathcal S$, an effective bound on the number of non-degenerate solutions of an $\mathcal S$-unit equation in $k$ variables over the rationals) and the other using only elementary methods. As a corollary, we find that, for fixed $c_1, x_1, \ldots,c_k, x_k \in \mathbf N^+$, the number of distinct prime factors of $c_1 x_1^n+\cdots+c_k x_k^n$ is bounded, as $n$ ranges over $\mathbf N^+$, if and only if $x_1=\cdots=x_k$.
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 $\prod_{i=1}^k \left(1-\frac{1}{φ(a_i)}\right)$, where $φ$ 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 $\prod_{i=1}^k \left(1-\frac{1}{a_i}\right)$.
Let $X$ be a first countable space which admits a non-trivial convergent sequence and let $\mathcal{I}$ be an analytic P-ideal. First, it is shown that the sets of $\mathcal{I}$-limit points of all sequences in $X$ are closed if and only if $\mathcal{I}$ is also an $F_σ$-ideal. Moreover, let $(x_n)$ be a sequence taking values in a Polish space without isolated points. It is known that the set $A$ of its statistical limit points is an $F_σ$-set, the set $B$ of its statistical cluster points is closed, and that the set $C$ of its ordinary limit points is closed, with $A\subseteq B\subseteq C$. It is proved the sets $A$ and $B$ own some additional relationship: indeed, the set $S$ of isolated points of $B$ is contained also in $A$. Conversely, if $A$ is an $F_σ$-set, $B$ is a closed set with a subset $S$ of isolated points such that $B\setminus S\neq \emptyset$ is regular closed, and $C$ is a closed set with $S\subseteq A\subseteq B\subseteq C$, then there exists a sequence $(x_n)$ for which: $A$ is the set of its statistical limit points, $B$ is the set of its statistical cluster points, and $C$ is the set of its ordinary limit points. Lastly, we discuss topological nature of the set of $\mathcal{I}$-limit points when $\mathcal{I}$ is neither $F_σ$- nor analytic P-ideal.
Let $D$ be a convex subset of a real vector space. It is shown that a radially lower semicontinuous function $f: D\to \mathbf{R}\cup \{+\infty\}$ is convex if and only if for all $x,y \in D$ there exists $α=α(x,y) \in (0,1)$ such that $f(αx+(1-α)y) \le αf(x)+(1-α)f(y)$.
Given a partition $\{I_1,\ldots,I_k\}$ of $\{1,\ldots,n\}$, let $(X_1,\ldots,X_n)$ be random vector with each $X_i$ taking values in an arbitrary measurable space $(S,\mathscr{S})$ such that their joint law is invariant under finite permutations of the indexes within each class $I_j$. Then, it is shown that this law has to be a signed mixture of independent laws and identically distributed within each class $I_j$. The representation is unique if and only if the set of these signed measures is weakly compact. We provide a necessary condition for the existence of a nonnegative directing measure. This is related to the notions of infinite extendibility and reinforcement. In the special case where $(X_1,\ldots,X_n)$ is an exchangeable sequence of $\{0,1\}$-valued random variables, the directing measure can be chosen nonnegative if and only if two effectively computable matrices are positive semi-definite.
Let $(X, \mathscr{L}, λ)$ and $(Y, \mathscr{M}, μ)$ be finite measure spaces for which there exist $A \in \mathscr{L}$ and $B \in \mathscr{M}$ with $0 < λ(A) < λ(X)$ and $0 < μ(B) < μ(Y)$, and let $I\subseteq \mathbf{R}$ be a non-empty interval. We prove that, if $f$ and $g$ are continuous bijections $I \to \mathbf{R}^+$, then the equation $$ f^{-1}\!\left(\int_X f\!\left(g^{-1}\!\left(\int_Y g \circ h\;dμ\right)\right)d λ\right)\! = g^{-1}\!\left(\int_Y g\!\left(f^{-1}\!\left(\int_X f \circ h\;dλ\right)\right)d μ\right)$$ is satisfied by every $\mathscr{L} \otimes \mathscr{M}$-measurable simple function $h: X \times Y \to I$ if and only if $f=c g$ for some $c \in \mathbf{R}^+$ (it is easy to see that the equation is well posed). An analogous, but essentially different, result, with $f$ and $g$ replaced by continuous injections $I \to \mathbf R$ and $λ(X)=μ(Y)=1$, was recently obtained in [Indag. Math. 27 (2016), 945-953].
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 $F_k$.
Catalan's conjecture claims that the Diophantine equation $x^p-y^q=1$ admits the unique solution $3^2-2^3=1$ in integers $x,y,p,q \ge 2$. The conjecture has been finally proved by P. Mihăilescu (2002) using the theory of cyclotomic fields and Galois modules. Here, relying only on elementary techniques, we prove several instances of this classical result. In particular, we prove the conjecture in the following cases: $p$ even (due to V.A. Lebesgue), $q$ is even (due to L. Euler and Chao Ko), $x$ divides $q$, $y$ divides $x-1$, $y$ is a power of a prime, and $y\le p^{p/2}$.
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 $μ^\ast: \mathcal{P}({\bf N})\to\bf R$ such that $μ^\ast({\bf N}) = 1$ and $μ^\ast(k \cdot X + h) = \frac{1}{k} μ^\ast(X)$ 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 Pólya, and upper analytic densities are examples of upper densities. We show that every upper density $μ^\ast$ 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 $μ^\ast$ is relaxed to the weaker condition that $μ^\ast(X) \le 1$ for every $X \subseteq \bf N$.
Let $R$ be a commutative ring, $f \in R[X_1,\ldots,X_k]$ a multivariate polynomial, and $G$ a finite subgroup of the group of units of $R$ satisfying a certain constraint, which always holds if $R$ is a field. Then, we evaluate $\sum f(x_1,\ldots,x_k)$, where the summation is taken over all pairwise distinct $x_1,\ldots,x_k \in G$. In particular, let $p^s$ be a power of an odd prime, $n$ a positive integer coprime with $p-1$, and $a_1,\ldots,a_k$ integers such that $φ(p^s)$ divides $a_1+\cdots+a_k$ and $p-1$ does not divide $\sum_{i \in I}a_i$ for all non-empty proper subsets $I\subseteq \{1,\ldots,k\}$; then $$ \sum x_1^{a_1}\cdots x_k^{a_k} \equiv \frac{φ(p^s)}{\mathrm{gcd}(n,φ(p^s))}(-1)^{k-1}(k-1)! \,\,\bmod{p^s}, $$ where the summation is taken over all pairwise distinct $n$-th residues $x_1,\ldots,x_k$ modulo $p^s$ coprime with $p$.
Let $n\ge 5$ be an odd integer. It is shown that $\{1^{σ(1)},\ldots,n^{σ(n)}\}$ is a complete residue system modulo $n$ for some permutation $σ$ of $\{1,\ldots,n\}$ if and only if $\frac{1}{2}(n-1)$ is a Sophie Germain prime. Partial results are obtained also for the case $n$ even.
Let $f$ and $g$ be real-valued continuous injections defined on a non-empty real interval $I$, and let $(X, \mathscr{L}, λ)$ and $(Y, \mathscr{M}, μ)$ be probability spaces in each of which there is at least one measurable set whose measure is strictly between $0$ and $1$. We say that $(f,g)$ is a $(λ, μ)$-switch if, for every $\mathscr{L} \otimes \mathscr{M}$-measurable function $h: X \times Y \to \mathbf{R}$ for which $h[X\times Y]$ is contained in a compact subset of $I$, it holds $$ f^{-1}\!\left(\int_X f\!\left(g^{-1}\!\left(\int_Y g \circ h\;dμ\right)\right)d λ\right)\! = g^{-1}\!\left(\int_Y g\!\left(f^{-1}\!\left(\int_X f \circ h\;dλ\right)\right)d μ\right)\!, $$ where $f^{-1}$ is the inverse of the corestriction of $f$ to $f[I]$, and similarly for $g^{-1}$. We prove that this notion is well-defined, by establishing that the above functional equation is well-posed (the equation can be interpreted as a permutation of generalized means and raised as a problem in the theory of decision making under uncertainty), and show that $(f,g)$ is a $(λ, μ)$-switch if and only if $f = ag + b$ for some $a,b \in \mathbf R$, $a \ne 0$.
For all integers $n \geq k \geq 1$, define $H(n,k) := \sum 1 / (i_1 \cdots i_k)$, where the sum is extended over all positive integers $i_1 < \cdots < i_k \leq n$. These quantities are closely related to the Stirling numbers of the first kind by the identity $H(n,k) = s(n + 1, k + 1) / n!$. Motivated by the works of Erdős-Niven and Chen-Tang, we study the $p$-adic valuation of $H(n,k)$. In particular, for any prime number $p$, integer $k \geq 2$, and $x \geq (k-1)p$, we prove that $ν_p(H(n,k)) < -(k - 1)(\log_p(n/(k - 1)) - 1)$ for all positive integers $n \in [(k-1)p, x]$ whose base $p$ representations start with the base $p$ representation of $k - 1$, but at most $3x^{0.835}$ exceptions. We also generalize a result of Lengyel by giving a description of $ν_2(H(n,2))$ in terms of an infinite binary sequence.
The aim of this note is to provide an upper bound of the number of positive integers $\le x$ which can be written as $φ(n)$ for some positive integer $n$, where $φ$ stands for the Euler's function. The order of magnitude of this estimate, which is roughly $x/\sqrt[4]{\ln x}$, implies that the set of Euler's values contains arbitrarily large gaps.