On the Monotonicity of Higher-Fold Representation Functions

For a positive integer $h$, let $R_{A,h}(n)$ denote the number of ordered representations $n=s_1+\cdots +s_h$ with all $s_i\in A$. Let $$B=\{0\}\cup \{m\ge 1:\text{ the base-4 expansion of }m\text{ begins with }1\text{ or }2\}\mathrm{.}$$ Shallit proved that $R_{B,3}(n)$ is strictly increasing, thereby disproving a 2002 conjecture of Dombi. In this paper, by using linear bounds for $R_{B,3}(n+1)-R_{B,3}(n)$ and a convolution argument, we prove the polynomial order of $R_{B,h}(n+1)-R_{B,h}(n)$ for every integer $h\ge 3$. More precisely, for every integer $h\ge 3$, there exist constants $c_h,C_h>0$, depending only on $h$, such that $$c_h{n}^{h-2}\le R_{B,h}(n+1)-R_{B,h}(n)\le C_h{n}^{h-2}$$ for all integers $n\ge 1$. We also construct a co-infinite set $C\subset \mathbb{N}$ satisfying ${\lim }_{n\to \mathrm{\infty }}C(n)\mathrm{/}n=1$ such that $R_{C,h}(n)$ is strictly increasing for every integer $h\ge 3$. This answers a problem of Dombi posed in 2002. We also pose some problems for further research.