Representation functions with prescribed rates of growth

Abstract

Fix an integer $h\ge 2$, and let $b_1,\dots ,b_h$ be (not necessarily distinct) positive integers with $\gcd (b_1,\dots ,b_h)=1$. For any subset $A\subseteq \mathbb{N}$, let $r_A(n)$ denote the number of solutions $(k_1,\dots ,k_h)\in A^h$ to the equation \[ b_1 k_1 + \cdots + b_h k_h = n. \] Given a function $F$ satisfying $F(n)\le r_{\mathbb{N}}(n)$, we ask: when does there exist a set $A\subseteq \mathbb{N}$ such that $r_A(n)\sim F(n)$? We prove that this is always possible when $F$ is regularly varying and satisfies ${\lim }_{n\to \infty }F(n)/\log n=\infty$. If one only requires $r_A(n)\asymp F(n)$, much weaker regularity conditions suffice: we show such a set $A$ exists for every increasing function $F$ satisfying $F(2x)\ll F(x)$ and $\log x\ll F(x)\ll x^{h-1}$. Finally, we give a probabilistic heuristic supporting the following: if $A\subseteq \mathbb{N}$ satisfies ${\mathrm{limsup}}_{n\to \infty }{r}_A(n)/\log n<1$, then $r_A(n)=0$ for infinitely many $n$.

arXiv comment28 pages. Reglued the ranges in Theorem 1.3 into a single full-range statement; minor exposition and notation edits