A Dombi Counterexample with Positive Lower Density

Let $r(k,A,n)$ denote the number of representations of $n$ as a sum of $k$ elements of a set $A\subseteq \mathbb{N}$. In 2002, Dombi conjectured that if $A$ is co-infinite, then the sequence $(r(k,A,n){)}_{n\ge 0}$ cannot be strictly increasing. Using tools from automata theory and logic, we give an explicit counterexample where $\mathbb{N}\setminus A$ has positive lower density.