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 $, such as a lower bound on its minimum, be helpful in establishing such an asymptotic result?
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3$\begingroup$ Obviously not: take $\Lambda_n=(n\mathbf{Z})\times\mathbf{Z}^{n-1\}$. Then $\Lambda_n\cap [0,2[^n$=\{0\}\times\{0,1\}^{n-1}$ has cardinal $2^{n-1}=\frac{2^n}{2}\gg \frac{2^n}{n}$. So you need additional conditions; maybe as you suggest with a lower bound on its minimum. $\endgroup$– YCorCommented May 30, 2020 at 23:28
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1$\begingroup$ To make the comment by YCor more precise, you need some control on the successive minima on your lattice. If it is too skew then what you're asking for won't work, because there is a lower rank sub-lattice that has too many points in the box. $\endgroup$– Stanley Yao XiaoCommented May 30, 2020 at 23:50
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$\begingroup$ Thank you both! Are you perhaps aware of a general result that bounds the number of lattice points inside a convex region in terms of successive minima? I know of Davenport's theorem (from the replies to related questions), but it does not seem to give sharp bounds for what I need. $\endgroup$– alephCommented May 31, 2020 at 7:46
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1 Answer
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The following answer is from A Reverse Minkowski Theorem. 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.
For any lattice $L \subseteq \mathbb{R}^n$ with $\det(L) \leq 1$ and $r > 0$, $$|L \cap rB_2^n| \geq 2^{-n}\mathsf{vol}(rB_2^n) = \frac{1}{\sqrt{\pi n}}\left(\frac{\pi e r^2}{2n}\right)^{n/2}(1 + o(1)).$$
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.
A lattice $L\subseteq \mathbb{R}^n$ is stable if $\det L = 1$, and $\det(L') \geq 1$ for all sublattices $L'\subseteq L$.
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 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.
