相关论文: Additive triples in groups of odd prime order

A subset $A$ of a group $G$ is called $(k, l)$-{\it sumset}, if $A= kB-lB$ for some $B\subseteq G$, where $kB-lB={x_1+...+x_k-x_{k+1}-...-x_{k+l} : x_1,..., x_{k+l}\in B}.$ Upper and lower bounds for the number $(k, l)$-sumsets in groups of…

In this paper, we consider some additive properties of integers with restricted digit expansions. Let $b\geq 3$ be an integer and $B_b$ be the set of integers whose base $b$ expansions have only digits $\{0,1\}.$ Let $a,b,c$ be three…

We prove character sum estimates for additive Bohr subsets modulo a prime. These estimates are analogous to classical character sum bounds of Polya-Vinogradov and Burgess. These estimates are applied to obtain results on recurrence mod $p$…

We show that there are $O(B^{3/5-3/1555+\ep})$ triples $(x,y,z)$ of square-full integesr up to $B$ satisfying the equation $x+y=z$ for any fixed $\ep>0$. This is the first improvement over the `easy' exponent $3/5$, given by Browning and…

We show that a dense subset of a sufficiently large group multiplication table contains either a large part of the addition table of the integers modulo some $k$, or the entire multiplication table of a certain large abelian group, as a…

Graham, R\"odl, and Ruci\'nski originally posed the problem of determining the minimum number of monochromatic Schur triples that must appear in any 2-coloring of the first $n$ integers. This question was subsequently resolved independently…

In this paper, we study additive properties of finite sets of lattice points on spheres in $3$ and $4$ dimensions. Thus, given $d,m \in \mathbb{N}$, let $A$ be a set of lattice points $(x_1, \dots, x_d) \in \mathbb{Z}^d$ satisfying $x_1^2 +…

Kemnitz Conjecture [9] states that if we take a sequence of elements in $Z_{p}^{2}$ of length $4p-3$, $p$ is a prime number, then it has a subsequence of length $p$, whose sum is $0$ modulo $p$. It is known that in $Z_{p}^{3}$ to get a…

We prove an asymptotic for the number of additive triples of bijections $\{1,\dots,n\}\to\mathbb{Z}/n\mathbb{Z}$, that is, the number of pairs of bijections $\pi_1,\pi_2\colon \{1,\dots,n\}\to\mathbb{Z}/n\mathbb{Z}$ such that the pointwise…

Let $n, k$ and $a$ be positive integers. The Stirling numbers of the first kind, denoted by $s(n,k)$, count the number of permutations of $n$ elements with $k$ disjoint cycles. Let $p$ be a prime. In recent years, Lengyel, Komatsu and…

Let $A$ and $B$ be sets of $k\ge5$ elements in $F=\mathbb{Z}/p\mathbb{Z}$ the field with $p>2k-2$ elements. We denote by $A\dot{+}B$ the set of different elements of $F$ that can be written in the form $a+b$, where $a\in A$, $b\in B$,…

We present an elementary proof that the Schur polynomial corresponding to an increasing sequence of exponents (c_0,..., c_{n-1}) with c_0 = 0 is irreducible over every field of characteristic p whenever the numbers d_i = c_{i+1} - c_i are…

In this paper, we study the theory of complements, introduced by Shokurov, for Calabi-Yau type varieties with the coefficient set $[0,1]$. We show that there exists a finite set of positive integers $\mathcal{N}$, such that if a threefold…

In the classical sense, the set B consists of all integers which can be written as a sum of two perfect squares. In other words, these are the values attained by norms of integral ideals over the Gaussian field Q(i). G.J. Rieger (1965) and…

Recently, Hajdu and S\'{a}rk\"{o}zy studied the multiplicative decompositions of polynomial sequences. In particular, they showed that when $k \geq 3$, each infinite subset of $\{x^k+1: x \in \mathbb{N}\}$ is multiplicatively irreducible.…

We call an integer a \emph{near-square} if its absolute value is a square or a prime times a square. We investigate such near-squares in the binary recurrence sequences defined for integers $a \geq 3$ by $u_{0}(a)=0$, $u_{1}(a)=1$ and…

It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime…

Let $G$ be a finite group. If $\Gamma$ is a permutation group with $G_{right}\leq\Gamma\leq Sym(G)$ and $\mathcal{S}$ is the set of orbits of the stabilizer of the identity $e=e_{G}$ in $\Gamma$, then the $\mathbb{Z}$-submodule…

Let S(n,k) denote the Stirling numbers of the second kind. We prove that the p-adic limit of S(p^e a + c, p^e b + d) as e goes to infinity exists for all integers a, b, c, and d. We call the limiting p-adic integer S(p^\infty a + c,…

A. Smoktunowicz and L. Vendramin conjectured that if $A=(A,\oplus,\odot)$ is a finite skew brace with solvable additive group $A_{\oplus}$, then the multiplicative group $A_{\odot}$ of $A$ is also solvable. Proving or disproving this…