AI 中文总结
本文解决了Sárközy和Sós在1997年提出的两个问题，一是关于特定条件下集合\(A\)补集的有限性问题，二是存在满足一定条件的算术函数\(f\)及集合\(A\)，使\(|r_1(A,n) - f(n)| = o((f(n))^{1/2})\)在特定序列上成立。
AI 中文摘要
对于集合\(A\subseteq\mathbb{N}_0\)，\(r_1(A,n)\)表示方程\(a + a^{\prime}=n\)（\(a,a^{\prime}\in A\)）的解的数量，\(r_2(A,n)\)表示满足\(a\leq a^{\prime}\)的此类解的数量，它们被称为加法表示函数。本文解决了Sárközy和Sós在1997年提出的两个问题。一是若\(A\)无限且对足够大的\(m\)有\(r_2(A,2m + 1)\geq r_2(A,2m)\)，则\(A\)的补集有限，这对文献中的问题3.1给出否定答案；二是存在算术函数\(f\)及集合\(A\)，使得在密度为\(1\)的整数序列\(n\)上\(|r_1(A,n) - f(n)| = o((f(n))^{1/2})\)，这对文献中的问题3.3给出肯定答案。
英文摘要
For a set $A\subseteq\mathbb{N}_0$, let $r_1(A,n)$ denote the number of solutions of the equation $a+a^{\prime}=n$ with $a,a^{\prime}\in A$, and let $r_2(A,n)$ denote the number of such solutions subject to $a\le a^{\prime}$. These functions are called additive representation functions (as first considered by Erdős, Sárközy and Sós). In this paper, we resolve two problems posed by Sárközy and Sós in 1997. First, if $A$ is infinite and $r_2(A,2m+1)\ge r_2(A,2m)$ for every sufficiently large $m$, then the complement of $A$ is finite. This gives a negative answer to Problem 3.1 in~\cite{SarkozySos1997}. Secondly, there exist an arithmetic function $f$ satisfying $f(n) \to \infty$, $f(n+1) \ge f(n)$ for $n > n_0$, and $f(n) = o\left(\frac{n}{(\log n)^2}\right)$, and a set $A$ such that \( |r_1(A,n) - f(n)| = o((f(n))^{1/2}) \) holds on a sequence of integers $n$ whose density is $1$. This gives a positive answer to Problem 3.3 in~\cite{SarkozySos1997}.
