Counterexamples to a Conjecture of Dombi in Additive Number Theory

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 $n\to |\{(a,b,c):n=a+b+c\text{ and }a,b,c\in A\}|$, counting the number of $3$-compositions using elements of $A$ only, is strictly increasing.

arXiv commentadditional author added; largely rewritten with different example