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

Let $\F$ be the finite field of odd prime power order $q$, We find explicit expressions for the number of triples $\{\al-1,\al,\al+1 \}$ of consecutive non-zero squares in $\F$ and similarly for the number of triples of consecutive…

We answer a question of Hegyv\'ari and Ruzsa concerning effective estimates of the Bohr-regularity of certain triple sums of sets with positive upper Banach densities in the integers. Our proof also works for any discrete amenable group,…

Let $A$ and $B$ be sets of nonnegative integers. For a positive integer $n$ let $R_{A}(n)$ denote the number of representations of $n$ as the sum of two terms from $A$. Let $\displaystyle s_{A}(x) = \max_{n \le x}R_{A}(n)$ and…

We give a new bound on the number of collinear triples for two arbitrary subsets of a finite field. This improves on existing results which rely on the Cauchy inequality. We then us this to provide a new bound on trilinear and quadrilinear…

We establish a sharp point-sphere incidence bound in finite fields for point sets exhibiting controlled additive structure. Working in the framework of \((4,s)\)-Salem sets, which quantify pseudorandomness via fourth-order additive energy,…

For an integer $t \geq 3$, let $\mathcal{L}(t)$ denote the linear equation $x_1 + x_2 + \cdots + x_{t-1} = x_t,$ where all variables are positive integers. For integers $k \geq 1$ and $t_0,t_1,\dots,t_{k-1} \geq 3$, the generalized Schur…

In this paper we first investigate for what positive integers $a,b,c$ every nonnegative integer $n$ can be represented as $x(ax+1)+y(by+1)+z(cz+1)$ with $x,y,z$ integers. We show that $(a,b,c)$ can be either of the following seven triples:…

The notions of special and extraspecial pairs of roots were introduced by Carter for calculating structure constants, [Ca72]. Let $\{r, s\}$ be a special pair of roots for which the structure constant $N(r,s)$ is sought, and let $\{r_1,…

The first open case of the Brown, Erd\H{o}s, S\'os conjecture is equivalent to the following; For every $c>0$ there is a threshold $n_0$ so that if a quasigroup has order $n\geq n_0$ then for every subset of triples of the form $(a,b,ab),$…

In this paper, we study sum formulas for Schur multiple zeta values and give a generalization of the sum formulas for multiple zeta(-star) values. We show that for ribbons of certain types, the sum over all admissible Young tableaux of this…

We deduce some elementary pairwise disjointness and semi-disjointness conditions on triples of subsets in arbitrary groups satisfying the so-called triple product property (TPP) as originally defined by H. Cohn and C. Umans in 2003. This…

Given a dense subset $A$ of the first $n$ positive integers, we provide a short proof showing that for $p=\omega(n^{-2/3})$ the so-called {\sl randomly perturbed} set $A \cup [n]_p$ a.a.s. has the property that any $2$-colouring of it has a…

We obtain a new bound on certain double sums of multiplicative characters improving the range of several previous results. This improvement comes from new bounds on the number of collinear triples in finite fields, which is a classical…

The undecidability of the additive theory of primes (with identity) as well as the theory Th(N,+, n -> p\_n), where p\_n denotes the (n+1)-th prime, are open questions. As a possible approach, we extend the latter theory by adding some…

Building on previous work by Lambert, Plagne and the third author, we study various aspects of the behavior of additive bases in infinite abelian groups and semigroups. We show that, for every infinite abelian group $T$, the number of…

The $3k-4$ Theorem asserts that, if $A,\,B\subseteq \mathbb Z$ are finite, nonempty subsets with $|A|\geq |B|$ and $|A+B|=|A|+|B|+r< |A|+2|B|-3$, then there are arithmetic progressions $P_A$ and $P_B$ of common difference with $X\subseteq…

It is the purpose of this thesis to enunciate and prove a collection of explicit results in the theory of prime numbers. First, the problem of primes in short intervals is considered. We prove that there is a prime between consecutive cubes…

We call a matrix blocky if, up to row and column permutations, it can be obtained from an identity matrix by repeatedly applying one of the following operations: duplicating a row, duplicating a column, or adding a zero row or column.…

Two infinite sequences A and B of non-negative integers are called additive complements, if their sum contains all sufficiently large integers. Let $A(x)$ and $B(x)$ be the counting functions of A and B. In this paper, we extend the results…

Let $A,B$ be sets of positive integers such that $A+B$ contains all but finitely many positive integers. S\'ark\"ozy and Szemer\'edi proved that if $ A(x)B(x)/x \to 1$, then $A(x)B(x)-x \to \infty $. Chen and Fang considerably improved…