Counterexamples to a Conjecture of Dombi in Additive Number Theory

TL;DR

This paper disproves Dombi's 2002 conjecture in additive number theory by constructing sets with infinite complements yet with a strictly increasing count of 3-sums, challenging previous assumptions.

Contribution

The authors provide explicit counterexamples to Dombi's conjecture, demonstrating that the conjecture does not hold in general.

Findings

Counterexamples with infinite complement sets

Sequence of 3-sums is strictly increasing

Disproves the conjecture of Dombi

Abstract

We disprove a 2002 conjecture of Dombi from additive number theory. More precisely, we find examples of sets $A \subset \mathbb{N}$ with the property that $\mathbb{N} \setminus A$ is infinite, but the sequence $n \rightarrow |\{ (a,b,c) \, : \, n=a+b+c \text{ and } a,b,c \in A \}|$, counting the number of $3$-compositions using elements of $A$ only, is strictly increasing.