Let $W\subseteq \mathbb Z$ be bounded below and normalised by $\inf W=1$, and put $\overline W=\mathbb Z_{>0}\setminus W$. Nathanson asked whether infinite sets of integers admit minimal additive complements. Chen and Yang [‘On a problem of Nathanson related to minimal additive complements’, SIAM J. Discrete Math. 26 (2012), 1532–1536] proved that the two-sided case is positive and that sufficiently large consecutive gaps in $\overline W$ force nonexistence in the one-sided case. Chen and Ding [‘On a problem of Nathanson on nonminimal additive complements’, Bull. Aust. Math. Soc. 114, 6–13] isolated a sparse-hole condition under which no ordinary minimal complement exists. We prove a finite-fold version of this sparse-hole obstruction. For an integer $h\geq 1$, write $hC=C+\cdots +C$. We show that if $\overline W$ contains a sequence $a_1<a_2<\cdots $ such that $a_{t+1}-a_t\to \infty $ and the four-neighbourhoods $(a_{t-2},a_{t+2})$ contain at most K elements of $\overline W$ for all sufficiently large t, then there is no set $C\subseteq \mathbb Z$ which is minimal subject to $hC+W=\mathbb Z$. This result recovers and broadens the Chen–Ding nonexistence theorem as the special case $h=1$.

Let A be a set of natural numbers. A set B of natural numbers is an additive complement of the set A if all sufficiently large natural numbers can be represented in the form $x+y$, where $x\in A$ and $y\in B$. We establish that if $A=\{a_i: i\in \mathbb {N}\}$ is a set of natural numbers such that $a_i<a_{i+1} $ for $i \in \mathbb {N}$ and $\liminf _{n\rightarrow \infty } (a_{n+1}/a_{n})>1$, then there exists a set $B\subset \mathbb {N}$ such that $B\cap A = \varnothing $ and B is a sparse additive complement of the set A.