方金辉: On generalized perfect difference sumsets

：方金辉 教授

Let $\mathbb{Z}$ be the set of integers and $\mathbb{N}$ be the set of positive integers. For a nonempty set $A$ of integers and any integers $n$, $h\ge 2$, denote $r_{A,h}(n)$ by the number of representations of $n$ of the form $n=a_1+a_2+\cdots+a_h$, where $a_1\le \cdots \le a_h$ and $a_i\in A$ for $i=1,2,\cdots,h$ and $d_{A}(n)$ by the number of $(a,a')$ with $a, a'\in A$ such that $n=a-a'$. The set $A$ of integers is called \emph{a perfect difference sumset} if $r_{A,2}(n)=1$ for all integers $n$ and $d_A(n)=1$ for all positive integers $n$. We considered generalized perfect difference sumsets and proved that, if two functions $f_1:\mathbb{N} \rightarrow \mathbb{N}$ and $f_2:\mathbb{Z} \rightarrow \mathbb{N}$ satisfy that $\liminf_{u \rightarrow \infty} f_1(u)\ge 2$ and $\liminf_{|u|\rightarrow \infty} f_2(u)\ge 2$, then there exists a set $A$ of integers such that: (i) $d_A(n)=f_1(n)$ for all $n\in \mathbb{N}$ and $r_{A,2}(n)=f_2(n)$ for all $n\in \mathbb{Z}$; (ii) $\limsup_{x\rightarrow\infty} A(-x,x)/\sqrt{x} \ge 1/\sqrt{2}$. Furthermore, following Cilleruelo and Nathanson's work, we proved that there exists a set $A$ of integers such that: (i) $r_{A,3}(n)=2$ for all $n\in \mathbb{Z}$ and $d_A(n)=1$ for all $n\in \mathbb{N}$; (ii) $A(x)\gg x^{\sqrt{5}-2+o(1)}$.