关于杨全会和陈永高提出的一个问题

On a Problem of Q. H. YANG and Y. G. CHEN

中文关键词:  表示函数  分拆  S\'ark\v{o}zy问题

英文关键词:representation function  partition  S\'{a}rk\"{o}zy problem

中文摘要:

      对任意的正整数与集合,令为解的个数.杨全会和陈永高证明了:若整数且,则不存在集合使得对所有充分大的整数成立,其中.对整数和,定义为满足对所有整数成立的集合的个数.杨全会和陈永高证明了是有限的,且.同时,他们问对任意整数,是否存在使得对所有整数成立.在本文中,我们给出了在时的准确公式.从而推出在时成立.

英文摘要:

      For any positive integers $k_1,k_2$ and any set $A\subseteq \mathbb{N}$, let $R_{k_1,k_2}(A,n)$ be the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$. Let $\bar{A}=\mathbb{N}\backslash A$. Yang and Chen proved that if $k_1$ and $k_2$ are two integers with $k_2>k_1\geq 2$ and $(k_1,k_2)=1$, then there does not exist any set $A\subseteq\mathbb{N}$ such that $R_{k_1,k_2}(A,n)=R_{k_1,k_2}(\bar{A},n)$ for all sufficiently large integers $n$. For two integers $k>1$ and $t\geq1$, define $f_{k}(t)$ to be the number of sets $A\subseteq\mathbb{N}$ such that $R_{1,k}(A,n)=R_{1,k}(\bar{A},n)$ holds for all integers $n\geq t$. Yang and Chen proved that $f_{k}(t)$ is finite and $\lim_{t\rightarrow\infty}\frac{\log f_{k}(t)}{t}=\log2$. They also asked if it is true that for any integers $k,l>1$ there exists $t_0(k,l)$ such that $f_k(t)=f_l(t)$ for all integers $t\geq t_0$. In this paper, we give the exact formula of $f_{k}(t)$ when $t\leq k$, which implies that $f_k(t)=f_l(t)$ for all integers $t\leq \min\{k,l\}$.