相关论文: On sums and products in a field

We improve a previous sum--products estimates in R, namely, we obtain that max{|A+A|,|AA|} \gg |A|^{4/3+c}, where c any number less than 5/9813. New lower bounds for sums of sets with small the product set are found. Also we prove some pure…

Let $\mathbb{F}_{q}$ be a finite field, $\mathbb{F}_{q^s}$ be an extension of $\mathbb{F}_q$, let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $n$ with $\gcd(n,q)=1$. We present a recursive formula for evaluating the exponential sum…

Let $\mathbb{F}_{q}$ be a finite field, $\mathbb{F}_{q^s}$ be an extension of $\mathbb{F}_q$, let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $n$ with $\gcd(n,q)=1$. We present a recursive formula for evaluating the exponential sum…

In the process of proving a sharpened form of G\r{a}rding's inequality, Fefferman & Phong demonstrated that every non-negative function $f\in C^{3,1}(\mathbb{R}^n)$ can be written as a finite sum of squares of functions in…

Let K be an infinite field such that its characteristic is not 2. We show that, for every $A\in\mathcal{M}_n(K)$ such that $\mathrm{rank}(A)\geq n/2$, there exists $B\in\mathcal{M}_n(K)$ such that $B$ is similar to $A$ and $A+B$ is…

Let $\mathbb{F}_q$ denote the finite field of odd characteristic $p$ with $q$ elements ($q=p^{n},n\in \mathbb{N} $) and $\mathbb{F}_q^*$ represent the nonzero elements of $\mathbb{F}_{q}$. In this paper, by using the Smith normal form we…

Let $\mathbb{F}_p$ be the field of a prime order $p.$ It is known that for any integer $N\in [1,p]$ one can construct a subset $A\subset\mathbb{F}_p$ with $|A|= N$ such that $$ \max\{|A+A|, |AA|\}\ll p^{1/2}|A|^{1/2}. $$ In the present…

Let $\mathbb{M}$ be the group of multiplicative characters of a finite field $\mathbb{F}$, and let $\mathbb{J}(\alpha, \beta)$ be the Jacobi sum, for $\alpha, \beta \in \mathbb{M}$. We observe that the function $\mathbb{J} \colon \mathbb{M}…

Let $k\ge 2$ be a fixed integer. We consider sums of type $\sum_{n_1^2+\cdots+ n_k^2\le x} F(n_1,\ldots,n_k)$, taken over the $k$-dimensional spherical region $\{(n_1,\ldots,n_k)\in {\Bbb Z}^k: n_1^2+\cdots+ n_k^2\le x\}$, where $F:{\Bbb…

Recently Ruckle \cite{RuckleArithmeticalSummability} introduced the theory of arithmetical summability suggested by the sum $ \sum_{k|m}f(k) $ as $ k $ ranges over the divisors of $m$ including $ 1 $ and $ m .$ Following Ruckle…

We prove that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1}=n_1$, and non-negative integers $j$ and $r$ with $j\leqslant m$, the following two expressions \begin{align*} &\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\brack…

It is proved that for any two subsets $A$ and $B$ of an arbitrary finite field $\Fq$ such that $|A||B|>q$ the identity $16AB=\Fq$ holds. Moreover, it is established that for every subsets $X, Y\subset \Fq$ with the property $|X||Y|\geqslant…

In this paper we study the factors of some alternating sums of products of binomial and q-binomial coefficients. We prove that for all positive integers n_1,...,n_m, n_{m+1}=n_1, and 0\leq j\leq m-1, {n_1+n_{m}\brack…

The main result of the paper is the following. Let a non-degenerate distribution have finite moments $\mu_k$ of all orders $k=0,1,2,\ldots$. Then the sequence $\{\mu_k/k!, \; k=0,1,2,\ldots\}$ either contains infinitely many different terms…

Let $E \subseteq \mathbb{F}_q^2$ be a set in the 2-dimensional vector space over a finite field with $q$ elements, which satisfies $|E| > q$. There exist $x,y \in E$ such that $|E \cdot (y-x)| > q/2.$ In particular, $(E+E) \cdot (E-E) =…

Let $k\ge 2$ be a fixed integer. We consider sums of type $\sum_{n_1\cdots n_k\le x} F(n_1,\ldots,n_k)$, taken over the hyperbolic region $\{(n_1,\ldots,n_k)\in {\Bbb N}^k: n_1\cdots n_k\le x\}$, where $F:{\Bbb N}^k\to {\Bbb C}$ is a given…

This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers $n$ such that $n, n+h$ and $n+k$ are all sums of two squares where $h$ and $k$…

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer…

We improve a result of Solymosi on sum-products in R, namely, we prove that max{|A+A|,|AA|}\gg |A|^{4/3+c}, where c>0 is an absolute constant. New lower bounds for sums of sets with small product set are found. Previous results are improved…

Let $p$ be a prime, $k$ a positive integer and let $\mathbb{F}_q$ be the finite field of $q=p^k$ elements. Let $f(x)$ be a polynomial over $\mathbb F_q$ and $a\in\mathbb F_q$. We denote by $N_{s}(f,a)$ the number of zeros of…