On Sárközy-Sós Theorem related to representation functions

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.