Related papers: On the additive bases problem in finite fields

Inspired by recent questions of Nathanson, we show that for any infinite abelian group $G$ and any integers $m_1, \ldots, m_H$, there exist finite subsets $A,B \subseteq G$ such that $|hA|-|hB|=m_h$ for each $1 \leq h \leq H$. We also…

We show the existence of an absolute constant $\alpha>0$ such that, for every $k \geq 3$, $G:=\mathop{\mathrm{Sym}}(k)$, and for every $H \leqslant G$ of index at least $3$, one has $|H/[H,H]| \leq |G:H|^{\alpha/ \log \log |G:H|}$. This…

We establish a link between abelian regular subgroup of the affine group, and commutative, associative algebra structures on the underlying vector space that are (Jacobson) radical rings. As an application, we show that if the underlying…

A set of quasi-uniform random variables $X_1,...,X_n$ may be generated from a finite group $G$ and $n$ of its subgroups, with the corresponding entropic vector depending on the subgroup structure of $G$. It is known that the set of entropic…

We first prove Bosch-L\"utkebohmert-Raynaud's conjectures on existence of global N\'eron models of not necessarily semi-abelian algebraic groups in the perfect residue fields case. We then give a counterexample to the existence in the…

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any…

Let G be a finite group and let p be a prime. A module for G over a field of characteristic p is called algebraic if it satisfies a polynomial, with addition and multiplication given by direct sum and tensor product. In some sense, having…

A finite group G is admissible over a field M if there is a division algebra whose center is M with a maximal subfield G-Galois over M. We consider nine possible notions of being admissible over M with respect to a subfield K of M, where…

We show that the change of basis matrices of a set of $m$ bases of a finite vector space is a connected groupoid of order $m^2$. We define a general method to express the elements of change of basis matrices as algebraic expressions using…

Let A be a subset of a finite abelian group G. We say that A is sum-free if there is no solution of the equation x + y = z, with x, y, z belonging to the set A. In this paper we shall characterise the largest possible sum-free subsets of G…

We obtain asymptotic formulas for the sums $\sum_{n_1,\ldots,n_k\le x} f((n_1,\ldots,n_k))$ and $ \sum_{n_1,\ldots,n_k\le x} f([n_1,\ldots,n_k])$ involving the gcd and lcm of the integers $n_1,\ldots,n_k$, where $f$ belongs to certain…

Let k be a perfect field and A a finite dimensional k-algebra of finite global dimension (e.g. the path algebra of a finite quiver without oriented cycles). Making use of the recent theory of noncommutative motives, we prove that the value…

Using Serre's adelic interpretation of cohomology, we develop a `differential and integral calculus' on an algebraic curve X over an algebraically closed filed k of constants of characteristic zero, define algebraic analogs of additive…

Let p be a prime number. Let G be a finite abelian p-group of exponent n (written additively) and A be a non-empty subset of $]n[:= \{1,2,..., n\}$ such that elements of A are incongruent modulo p and non-zero modulo p. Let $k \geq…

It was conjectured by Jaeger, Linial, Payan, and Tarsi in 1992 that for any prime number $p$, there is a constant $c$ such that for any $n$, the union (with repetition) of the vectors of any family of $c$ linear bases of $\mathbb{Z}_p^n$…

Let $k$ be a field, $G$ be an abelian group and $r\in \mathbb N$. Let $L$ be an infinite dimensional $k$-vector space. For any $m\in End_k(L)$ we denote by $r(m)\in [0,\infty ]$ the rank of $m$. We define by $R(G,r,k)\in [0,\infty]$ the…

Let K be a field of characteristic 0 and let n be a natural number. Let Gamma be a subgroup of the multiplicative group $(K^\ast)^n$ of finite rank r. Given $A_2,...,a_n\in K^\ast$ write $A(a_1,...,a_n,\Gamma)$ for the number of solutions…

A finite abelian group $G$ of cardinality $n$ is said to be of type III if every prime divisor of $n$ is congruent to 1 modulo 3. We obtain a classification theorem for sum-free subsets of largest possible cardinality in a finite abelian…

We prove some improvements of the classical Weil bound for one variable additive and multiplicative character sums associated to a polynomial over a finite field $k=\Fq$ for two classes of polynomials which are invariant under a large…

We prove that for sets $A, B, C \subset \mathbb{F}_p$ with $|A|=|B|=|C| \leq \sqrt{p}$ and a fixed $0 \neq d \in \mathbb{F}_p$ holds $$ \max(|AB|, |(A+d)C|) \gg|A|^{1+1/26}. $$ In particular, $$ |A(A+1)| \gg |A|^{1 + 1/26} $$ and $$…