Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. The representation function $R_{ A,k}(n)$ for the set $A$ is the number of representations of a nonnegative integer $n$ as the sum of $k$ terms from $A$. Let $A(n)$ denote the counting function of $A$.Bell and Shallit recently gave a counterexample for a conjecture of Dombi and proved that if $A(n)=o(n^{ \frac{ k-2}{ k}-\epsilon})$ for some $\epsilon>0$, then $R_{ \mathbb{ N}\setminus A,k}(n)$ is eventually strictly increasing. In this paper, we improve this result to $A(n)=O(n^{ \frac{ k-2}{ k-1}})$. We also give an example to show that this bound is best possible.
Categories: math.NT math.CO