Restricted partition functions and additive complements

Abstract

Let $\mathbb{N}$ be the set of positive integers. For subsets $\mathcal{A},\mathcal{M}\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $p(n,\mathcal{A},\mathcal{M})$ denote the number of representations of $n$ in the form $$n=\sum _{a\in \mathcal{A}}{m}_{a}a,$$ where $m_a\in \mathcal{M}\cup \{0\}$ for all $a\in \mathcal{A}$, and only finitely many $m_a$ are nonzero. We prove that there exist two infinite sets $\mathcal{A}=\{a_n{\}}_{n=1}^{\infty }$ and $\mathcal{M}$ of positive integers such that $$\underset{n\to \infty }{\lim }\frac{\log a_{n+1}-\log a_n}{\log n}=+\infty ,$$ $p(n,\mathcal{A},\mathcal{M})>0$ for every $n\in \mathbb{N}$, and $p$ has polynomial growth. More generally, we prove a construction that associates restricted partition functions of polynomial growth with additive complements satisfying a simple counting condition. This answers a 2016 question of Dai and Chen in the affirmative.

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