arXiv subjects

Salvatore Tringali

Publications and source records attributed to Salvatore Tringali.

47 records · Page 3Linked to original sources

arXiv2016

Cauchy-Davenport type inequalities, I

Let $\mathbb G = (G, +)$ be a group (either abelian or not). Given $X, Y \subseteq G$, we denote by $\langle Y \rangle$ the subsemigroup of $\mathbb G$ generated by $Y$, and we set $$γ(Y) := \sup_{y_0 \in Y} \inf_{y_0 \ne y \in Y} {\rm ord}(y - y_0)$$ if $|Y| \ge 2$ and $γ(Y) := |Y|$ otherwise. We prove that if $\langle Y \rangle$ is commutative, $Y$ is non-empty, and $X+2Y \neq X + Y + y$ for some $y \in Y$, then $$ |X+Y| \ge |X|+\min(γ(Y), |Y| - 1). $$ Actually, this is obtained from a more general result, which improves on previous work of the author on sumsets in cancellative semigroups, and yields a comprehensive generalization, and in some cases a considerable strengthening, of various additive theorems, notably including the Chowla-Pillai theorem (on sumsets in finite cyclic groups) and the specialization to abelian groups of the Hamidoune-Shatrowsky theorem.

math.CO↗

arXiv2016

A unified approach to the theory of normed structures - Part I: The single-sorted case

We introduce the concept of a prenormed model of a particular kind of finitary single-sorted first-order theories, interpreted over a category with finite products. These are referred to as prealgebraic theories, for the fact that their signature comprises, together with arbitrary function symbols (of finite ariety), only relation symbols whose interpretation, in any possible model, is a reflexive and transitive binary relation, namely a preorder. The result is an abstract approach to the very concept of norm and, consequently, to the theory of normed structures.

math.CT↗

arXiv2016

Plots and Their Applications - Part I: Foundations

The primary goal of this paper is to abstract notions, results and constructions from the theory of categories to the broader setting of plots. Loosely speaking, a plot can be thought of as a non-associative non-unital category with a "relaxed" composition law: Besides categories, this includes as a special case graphs and neocategories in the sense of Ehresmann, Gabriel's quivers, Mitchell's semicategories, and composition graphs, precategories and semicategories in the sense of Schröder. Among other things, we formulate an "identity-free" definition of isomorphisms, equivalences, and limits, for which we introduce regular representations, punctors, $\mathcal M$-connections, and $\mathcal M$-factorizations. Part of the material will be used in subsequent work to lay the foundation for an abstract theory of "normed structures" serving as a unifying framework for the development of fundamental aspects of the theory of normed spaces, normed groups, etc., on the one hand, and measure spaces, perhaps surprisingly, on the other.

math.CT↗

arXiv2015

Generalized Goncarov polynomials

We introduce the sequence of generalized Gončarov polynomials, which is a basis for the solutions to the Gončarov interpolation problem with respect to a delta operator. Explicitly, a generalized Gončarov basis is a sequence $(t_n(x))_{n \ge 0}$ of polynomials defined by the biorthogonality relation $\varepsilon_{z_i}(\mathfrak d^{i}(t_n(x))) = n! \;\! δ_{i,n}$ for all $i,n \in \mathbf N$, where $\mathfrak d$ is a delta operator, $\mathcal Z = (z_i)_{i \ge 0}$ a sequence of scalars, and $\varepsilon_{z_i}$ the evaluation at $z_i$. We present algebraic and analytic properties of generalized Gončarov polynomials and show that such polynomial sequences provide a natural algebraic tool for enumerating combinatorial structures with a linear constraint on their order statistics.

math.CO↗

arXiv2015

Sums of dilates in ordered groups

We address the "sums of dilates problem" by looking for non-trivial lower bounds on sumsets of the form $k \cdot X + l \cdot X$, where $k$ and $l$ are non-zero integers and $X$ is a subset of a possibly non-abelian group $G$ (written additively). In particular, we investigate the extension of some results so far known only for the integers to the context of torsion-free or linearly orderable groups, either abelian or not.

math.CO↗

arXiv2015

The Davenport constant of a box

Given an additively written abelian group $G$ and a set $X\subseteq G$, we let $\mathscr{B}(X)$ denote the monoid of zero-sum sequences over $X$ and $\mathsf{D}(X)$ the Davenport constant of $\mathscr{B}(X)$, namely the supremum of the positive integers $n$ for which there exists a sequence $x_1 \cdots x_n$ of $\mathscr{B}(X)$ such that $\sum_{i \in I} x_i \ne 0$ for each non-empty proper subset $I$ of $\{1, \ldots, n\}$. In this paper, we mainly investigate the case when $G$ is a power of $\mathbb{Z}$ and $X$ is a box (i.e., a product of intervals of $G$). Some mixed sets (e.g., the product of a group by a box) are studied too, and some inverse results are obtained.

math.CO↗

arXiv2015

Cauchy-Davenport type theorems for semigroups

Let $\mathbb{A} = (A, +)$ be a (possibly non-commutative) semigroup. For $Z \subseteq A$ we define $Z^\times := Z \cap \mathbb A^\times$, where $\mathbb A^\times$ is the set of the units of $\mathbb{A}$, and $$γ(Z) := \sup_{z_0 \in Z^\times} \inf_{z_0 \ne z \in Z} {\rm ord}(z - z_0).$$ The paper investigates some properties of $γ(\cdot)$ and shows the following extension of the Cauchy-Davenport theorem: If $\mathbb A$ is cancellative and $X, Y \subseteq A$, then $$|X+Y| \ge \min(γ(X+Y),|X| + |Y| - 1).$$ This implies a generalization of Kemperman's inequality for torsion-free groups and strengthens another extension of the Cauchy-Davenport theorem, where $\mathbb{A}$ is a group and $γ(X+Y)$ in the above is replaced by the infimum of $|S|$ as $S$ ranges over the non-trivial subgroups of $\mathbb{A}$ (Hamidoune-Károlyi theorem).

math.GR↗

arXiv2014

On a system of equations with primes

Given an integer $n \ge 3$, let $u_1, \ldots, u_n$ be pairwise coprime integers $\ge 2$, $\mathcal D$ a family of nonempty proper subsets of $\{1, \ldots, n\}$ with "enough" elements, and $\varepsilon$ a function $ \mathcal D \to \{\pm 1\}$. Does there exist at least one prime $q$ such that $q$ divides $\prod_{i \in I} u_i - \varepsilon(I)$ for some $I \in \mathcal D$, but it does not divide $u_1 \cdots u_n$? We answer this question in the positive when the $u_i$ are prime powers and $\varepsilon$ and $\mathcal D$ are subjected to certain restrictions. We use the result to prove that, if $\varepsilon_0 \in \{\pm 1\}$ and $A$ is a set of three or more primes that contains all prime divisors of any number of the form $\prod_{p \in B} p - \varepsilon_0$ for which $B$ is a finite nonempty proper subset of $A$, then $A$ contains all the primes.

math.NT↗

arXiv2014

Small doubling in ordered semigroups

Let $\mathbb{A} = (A, \cdot)$ be a semigroup. We generalize some recent results by G. A. Freiman, M. Herzog and coauthors on the structure theory of set addition from the context of linearly orderable groups to linearly orderable semigroups, where we say that $\mathbb{A}$ is linearly orderable if there exists a total order $\le$ on $A$ such that $xz < yz$ and $zx < zy$ for all $x,y,z \in A$ with $x < y$. In particular, we find that if $S$ is a finite subset of $A$ generating a non-abelian subsemigroup of $\mathbb{A}$, then $|S^2| \ge 3|S|-2$. On the road to this goal, we also prove a number of subsidiary results, and most notably that for $S$ a finite subset of $A$ the commutator and the normalizer of $S$ are equal to each other.

math.CO↗

arXiv2013

A Cauchy-Davenport theorem for semigroups

We generalize the Davenport transform and use it to prove that, for a (possibly non-commutative) cancellative semigroup $\mathbb A = (A, +)$ and non-empty subsets $X,Y$ of $A$ such that the subsemigroup generated by $Y$ is commutative, we have $|X + Y| \ge \min(ω(Y), |X| + |Y| - 1)$, where $ω(Y) := \sup_{y_0 \in Y \cap \mathbb A^{\times}} \inf_{y \in Y \setminus \{y_0\}} | |$. This carries over the Cauchy-Davenport theorem to the broader setting of semigroups, and it implies, in particular, an extension of I. Chowla's and S.S. Pillai's theorems for cyclic groups and a notable strengthening of another generalization of the same Cauchy-Davenport theorem to commutative groups, where $ω(Y)$ in the above is replaced by the minimal order of the non-trivial subgroups of $\mathbb A$.

math.CO↗

arXiv2013

On the divisibility of $a^n \pm b^n$ by powers of $n$

We determine all triples $(a,b,n)$ of positive integers such that $a$ and $b$ are relatively prime and $n^k$ divides $a^n + b^n$ (respectively, $a^n - b^n$), when $k$ is the maximum of $a$ and $b$ (in fact, we answer a slightly more general question). As a by-product, it is found that, for $m, n \in \mathbb N^+$ with $n \ge 2$, $n^m$ divides $ m^n + 1$ if and only if $(m,n)=(2,3)$ or $(1,2)$, which generalizes problems from the 1990 and 1999 editions of the International Mathematical Olympiad. The results are related to a conjecture by K. Győry and C. Smyth on the finiteness of the sets $R_k^\pm(a,b) := \{n \in \mathbb N^+: n^k \mid a^n \pm b^n\}$, when $a,b,k$ are fixed integers with $k \ge 3$, $\gcd(a,b) = 1$ and $|ab| \ge 2$.

math.NT↗