  The $k$-tiling problem for a convex polytope $P$ is the problem of covering
$\mathbb R^d$ with translates of $P$ using a discrete multiset $\Lambda$ of
translation vectors, such that every point in $\mathbb R^d$ is covered exactly
$k$ times, except possibly for the boundary of $P$ and its translates. A
classical result in the study of tiling problems is a theorem of McMullen that
a convex polytope $P$ that 1-tiles $\mathbb R^d$ with a discrete multiset
$\Lambda$ can, in fact, 1-tile $\mathbb R^d$ with a lattice $\mathcal L$. A
generalization of McMullen's theorem for $k$-tiling was conjectured by Gravin,
Robins, and Shiryaev, which states that if $P$ $k$-tiles $\mathbb R^d$ with a
discrete multiset $\Lambda$, then $P$ $m$-tiles $\mathbb R^d$ with a lattice
$\mathcal L$ for some $m$. In this paper, we consider the case when $P$
$k$-tiles $\mathbb R^d$ with a discrete multiset $\Lambda$ such that every
element of $\Lambda$ is contained in a quasi-periodic set $\mathcal Q$ (i.e. a
finite union of translated lattices). This is motivated by the result of
Gravin, Kolountzakis, Robins, and Shiryaev, showing that for $d \in \{2,3\}$,
if a polytope $P$ $k$-tiles $\mathbb R^d$ with a discrete multiset $\Lambda$,
then $P$ $m$-tiles $\mathbb R^d$ with a quasi-periodic set $\mathcal Q$ for
some $m$. Here we show for all values of $d$ that if a polytope $P$ $k$-tiles
$\mathbb R^d$ with a discrete multiset $\Lambda$ that is contained in a
quasi-periodic set $\mathcal Q$ that satisfies a mild hypothesis, then $P$
$m$-tiles $\mathbb R^d$ with a lattice $\mathcal L$ for some $m$. This
strengthens the results of Gravin, Kolountzakis, Robins, and Shiryaev, and is a
step in the direction of proving the conjecture of Gravin et al.
