On S\'ark\"ozy-S\'{o}s Theorem related to representation functions

Abstract

Let $\mathbb{N}_0$ be the set of all nonnegative integers. For a nonempty set $\mathcal{A}\subseteq \mathbb{N}_0$ and integers $n,h\ge 2$, let $r_{h}(\mathcal{A},n)$ be the number of representations of $n$ as $a_1+\cdots+a_h$, where $a_1\le \cdots\le a_h$ and $a_i\in \mathcal{A}$ for $i=1,\cdots,h$. In 2016, Chen and Tang showed that, for any given distinct positive integers $u_1,\cdots,u_k$ and positive rational numbers $\alpha_1,\cdots,\alpha_k$ with $\alpha_1+\cdots+\alpha_k=1$, there are infinitely many sets $\mathcal{A}\subseteq \mathbb{N}_0$ such that $r_{h}(\mathcal{A},n)\ge 1$ for all nonnegative integers $n$ and the set of $n$ with $r_{h}(\mathcal{A},n)=u_i$ has density $\alpha_i$ for all integer $i=1,\cdots,k$. In this paper, we consider the irrational numbers $\alpha_i$ as well. As a main result, we prove that, for any nonnegative numbers $\alpha_0,\cdots,\alpha_m$ with $\alpha_0+\cdots+\alpha_m=1$, there are infinitely many sets $\mathcal{A}\subseteq \mathbb{N}_0$ such that the set of $n$ with $r_{2}(\mathcal{A},n)=i$ has density $\alpha_i$ for all integer $i=0,\cdots,m$. Other related results are also contained.