Let $\Omega \subset \mathbb{R}^n$ be open and bounded, and let $Q \subset \Omega$ be a hypercube. Furthermore, denote by $D$ the $n$-dimensional unit cube $(0,1)^n$. Let $k \in \mathbb{N}$ be big, i.e., $k \gg 1$. My question is the following:
How many translates of $D$ are completely contained in $k Q$?
In the paper I am reading the author claims that this number is equal to: \begin{equation} k^n \frac{\lvert Q\rvert}{\lvert D \rvert} + \mathcal{O}(k^{n-1}). \end{equation} However, I cannot prove this claim. For me it seems reasonable that the answer is something like $ck^n$ for some constant $c > 0$. But I have no idea how to prove the claim; in particular, I do not know how to compute this constant $c$. Moreover, I have no idea why the author needs the second term $\mathcal{O}(k^{n-1})$. I would be grateful for any suggestions. Thanks!