Let $ (\Lambda\_n) $ be a family of lattices, $ \Lambda\_n \subset \mathbb{Z}^n $, with $ \det\Lambda\_n \sim n $ as $ n \to \infty $ (meaning $ \lim\_{n\to\infty} n^{-1} \det\Lambda\_n = 1$). I am interested in the asymptotics of the number of points of $ \Lambda\_n $ in the hypercube $ [0,2)^n $. In particular, is it true that: $$ |\Lambda\_n \cap [0,2)^n| = |\Lambda\_n \cap \{0,1\}^n| \sim \frac{2^n}{n} $$ as $ n \to \infty $? Does someone know how to prove this type of statement? Would an additional condition on $ \Lambda\_n $, such as a lower bound on its minimum, be helpful in establishing such an asymptotic result?



Lattice points in hypercubes
The following answer is from [A Reverse Minkowski Theorem](https://arxiv.org/pdf/1611.05979.pdf). It deals with a sphere, rather than a hypercube. I am unaware of extensions to hypercubes, but the lower bound I will quote from it at least seems non-trivial, and of interest.


Minkowski's first theorem has a "point-counting" version, due to Blichfeldt and van der Corput.



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> For any lattice $L \subseteq \mathbb{R}^n$ with $\det(L) \leq 1$ and $r > 0$,
> $$|L \cap rB\_2^n| \geq 2^{-n}\mathsf{vol}(rB\_2^n) = \frac{1}{\sqrt{\pi n}}\left(\frac{\pi e r^2}{2n}\right)^{n/2}(1 + o(1)).$$
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This result is apparently given for arbitrary norms, so look at the linked paper for the proper citation/statement for $\ell\_\infty$.


Of course, to get an asymptotic, one needs an upper bound as well.
As the comments mention, one needs to rule out "skew" lattices (say of the form $t\mathbb{Z}\oplus \frac{1}{t}\mathbb{Z}$) somehow. The authors of the linked paper do this via the notion of *stable lattices*.



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> A lattice $L\subseteq \mathbb{R}^n$ is *stable* if $\det L = 1$, and $\det(L') \geq 1$ for all sublattices $L'\subseteq L$.
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They then give upper bounds for a number of point-counting contexts (assuming the underlying lattice is stable) in corollary 1.4. I will not copy them all here (there are 3 cases depending on what the particular value of $r$ one has). None of them appear to obviously (asymptotically) match the above upper bound (and in [later work](https://arxiv.org/pdf/2201.05201.pdf) on a similar problem, but without an obvious point-counting application, the authors mention the work I describe above is non-tight).


This is answer is mostly to say that there is recent work towards establishing these kinds of bounds, but I do not believe the hypercube has been examined, nor do I believe tight point-counting asymptotics have been established.
The assumption authors have found useful to get rid of the "skew" lattices described is known as *stability*, and has been used in other contexts going back a few decades.


