A Dombi Counterexample with Positive Lower Density

TL;DR

This paper provides a counterexample to Dombi's conjecture, demonstrating that a co-infinite set with positive lower density can have a strictly increasing sequence of representation counts, using automata theory and logic.

Contribution

The paper constructs an explicit counterexample to Dombi's conjecture, showing that positive lower density sets can have strictly increasing representation sequences.

Findings

Counterexample set with positive lower density exists

Sequence of representation counts can be strictly increasing

Automata theory and logic are used to construct the counterexample

Abstract

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 $(r(k,A,n))_{n \geq 0}$ 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.