Representation functions with prescribed rates of growth

TL;DR

This paper investigates the existence of subsets of natural numbers with prescribed growth rates of their representation functions, establishing conditions under which such sets can be constructed.

Contribution

It provides new results on constructing sets with representation functions matching given regularly varying or increasing functions, under specific growth conditions.

Findings

Existence of sets with $r_A(n) hicksim F(n)$ for regularly varying $F$ with $ o ext{infinity}$ over $rac{ ext{log } n}$.

Construction of sets with $r_A(n) hicksim F(n)$ for increasing $F$ satisfying certain regularity conditions.

Heuristic argument suggesting $r_A(n)/ ext{log } n$ cannot stay below 1 infinitely often.

Abstract

Fix an integer $h \geq 2$, and let $b_1, \ldots, b_h$ be (not necessarily distinct) positive integers with $\gcd(b_1, \ldots, b_h) = 1$. For any subset $A \subseteq \mathbb{N}$, let $r_A(n)$ denote the number of solutions $(k_1, \ldots, k_h) \in A^h$ to the equation \[ b_1 k_1 + \cdots + b_h k_h = n. \] Given a function $F$ satisfying $F(n) \leq r_{\mathbb{N}}(n)$, 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 $\limsup_{n\to\infty} r_A(n)/\log n < 1$, then $r_A(n) = 0$ for infinitely many $n$.