Related papers: On sets with sum and difference structure

For a set of nonnegative integers $A$, denote by $R_{A}(n)$ the number of unordered representations of the integer $n$ as the sum of two different terms from $A$. In this paper we partially describe the structure of the sets, which have…

Let $\mathbb{N}$ be the set of all nonnegative integers. For any integer $r$ and $m$, let $r+m\mathbb{N}=\{r+mk: k\in\mathbb{N}\}$. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_{S}(n)$ denote the number of solutions of the…

Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let the representation function $R_{S}(n)$ denote the number of solutions of the equation $n=s+s'$ with $s, s'\in S$ and $s<s'$. In…

Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_S(n)$ denote the number of solutions of the equation $n=s+s'$, $s, s'\in S$, $s<s'$. In this paper, we determine the…

Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_S(n)$ denote the number of solutions of the equation $n=s_1+s_2$, $s_1,s_2\in S$ and $s_1<s_2$. Let $A$ be the set of all…

We present a variety of new results on finite sets A of integers for which the sumset A+A is larger than the difference set A-A, so-called MSTD (more sums than differences) sets. First we show that there is, up to affine transformation, a…

For a set $A$ of nonnegative integers, let $R_2(A,n)$ and $R_3(A,n)$ denote the number of solutions to $n=a+a'$ with $a,a'\in A$, $a<a'$ and $a\leq a'$, respectively. In this paper, we prove that, if $A\subseteq \mathbb{N}$ and $N$ is a…

For two sets $A$ and $M$ of positive integers and for a positive integer $n$, let $p(n,A,M)$ denote the number of partitions of $n$ with parts in $A$ and multiplicities in $M$, that is, the number of representations of $n$ in the form…

Two sets of nonnegative integers $A=\{a_1<a_2<\cdots\}$ and $B=\{b_1<b_2<\cdots\}$ are defined as \emph{disjoint}, if $\{A-A\}\bigcap\{B-B\}=\{0\}$, namely, the equation $a_i+b_t=a_j+b_k$ has only trivial solution. In 1984, Erd\H os and…

Let $\mathcal{R}$ denote the set of integers $n$ that can be represented as the sum $n = x^2 + y^2$ with $(x,y) = 1$. Let $a$ and $b$ be integers with $a>0$, $a \nmid b$. We show that for sufficiently large positive integer $N$ there are…

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and let $t(a,b,c,d;n)$ be the…

Let $d(\cdot)$ denote the natural density on the positive integers. We characterize all sets $A,B$ with positive density satisfying $d(A+B)=d(A)+d(B)$, under the assumption that the two sets are not both contained in a proper finite union…

We show that for any set $A \subset \mathbb{N}$ with positive upper density and any $\ell,m \in \mathbb{N}$, there exist an infinite set $B\subset \mathbb{N}$ and some $t\in \mathbb{N}$ so that $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\…

A set of integers is called sum-free if it contains no triple $(x,y,z)$ of not necessarily distinct elements with $x+y=z$. In this paper, we provide a structural characterisation of sum-free subsets of $\{1,2,\ldots,n\}$ of density at least…

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. For a $k$-tuple of positive integers $\underline{\lambda} = (\lambda_{1}, \dots{} ,\lambda_{k})$ with $1 \le \lambda_{1} < \lambda_{2} < \dots{} < \lambda_{k}$, we…

A set A of positive integers is called a perfect difference set if every nonzero integer has an unique representation as the difference of two elements of A. We construct dense perfect difference sets from dense Sidon sets. As a consequence…

We prove that for any positive integer c there are at least N(c), $1\leq N(c) < \phi(c)/2$ representations of c as a sum of two positive integers a, b, with no common divisor, such that the N(c) radicals R(abc) are all greater than kc,…

In this paper we show that every set $A \subset \mathbb{N}$ with positive density contains $B+C$ for some pair $B,C$ of infinite subsets of $\mathbb{N}$, settling a conjecture of Erd\H{o}s. The proof features two different decompositions of…

Let $A$ be a finite multiset of powers of a positive integer $d>1$. We describe the structure of the set $\mathrm{span}(A)$ of all sums of submultisets of $A$, and in particular give a criterion of $\mathrm{span}(A)=\mathrm{span}(B)$ for…

Let $A\subseteq \mathbb{Z}_{\geq 0}$ be a finite set with minimum element $0$, maximum element $m$, and $\ell$ elements strictly in between. Write $(hA)^{(t)}$ for the set of integers that can be written in at least $t$ ways as a sum of $h$…