Minimal additive complements for $\{m,h\}$-sets with $h\equiv 1\ (\mathrm{mod}\ m)$
Acta Arithmetica
MSC: Primary 11B13
DOI: 10.4064/aa251120-28-2
Published online: 21 July 2026
Abstract
Let $C$ and $W$ be two sets of integers. If $C+W=\mathbb {Z}$, then we say that $C$ is an additive complement to $W$. If no proper subset of $C$ is an additive complement to $W$, then we say that $C$ is a minimal additive complement to $W$. Let $m$ and $h$ be two positive integers with $m\geq 2$, $h\equiv 1\pmod {m}$. We show that there exists an infinite, not eventually periodic set $W=\{w_i\}_{i=1}^{\infty }$ such that $w_{i+1}-w_i\in \{m,h\}$ for all $i$ and $W$ admits a minimal additive complement.
