Representation functions with prescribed rates of growth

Abstract: Fix $h\geq 2$, and let $b_1$ , $\ldots$, $b_h$ be not necessarily distinct positive integers with $\gcd = 1$. For any subset $A\subseteq \mathbb{N}$, write $r_{A}(n)$ ($n\geq 0$) for the number of solutions $(k_1,\ldots,k_h)\in A^h$ to [ b_1 k_1 + \cdots + b_h k_h = n. ] For which functions $F$ with $F(n) \leq r_{\mathbb{N}}(n)$ can we find $A\subseteq \mathbb{N}$ such that $r_{A}(n)\sim F(n)$? We show that if $F$ is regularly varying, then there exists such an $A$ provided $\lim\limits_{n\to\infty} F(n)/\log n = \infty$. If one only requires $r_{A}(n) \asymp F(n)$, we show there exists such an $A$ for every increasing $F$ satisfying $F(2x) \ll F(x)$ and $\liminf\limits_{n\to\infty} F(n)/\log n > 0$. Lastly, we give a probabilistic heuristic for the following statement: if $A\subseteq \mathbb{N}$ is such that $\limsup\limits_{n\to\infty} r_{A}(n)/\log n < 1$, then $r_{A}(n) = 0$ for infinitely many $n$.