On a problem of Ma and Chen concerning minimal additive complements

Source abstract

Let $m\geq1$ be an integer and let $W=(m\mathbb N_0+X)\cup U\cup V\subseteq\mathbb Z$ be bounded below, where $X\ne\varnothing$, $[U]_m\subseteq[X]_m$, $[X]_m\cap[V]_m=\varnothing$, and $V$ contains no infinite arithmetic progression with common difference $m$. Here $[S]_m$ denotes the set of residue classes modulo $m$ represented by $S$. Suppose that $W$ satisfies the $\mathcal C$-condition modulo $m$. For $0\leq r<m$, let $V_r=V\cap(r+m\mathbb Z)$, and put $V'=\bigcup_{|V_r|=\infty}V_r$, where the union is over $0\leq r<m$. We prove that $W$ has a minimal additive complement if $V'$ satisfies the big-hole property modulo $m$. This answers a problem posed by Ma and Chen.