Neukirch, Problem 4.4.2

Question

After proving the following (Minkowski's Theorem):

If $\Gamma$ is a complete lattice in the Euclidean vector space $V$ and $X$ is a centrally symmetric, convex subset of $V$ such that vol($X$) $>$ $2^{n}$vol($\Gamma$), then $X$ contains at least one nonzero lattice point $\gamma \in \Gamma$.

Neukirch asks the question regarding the fact that this is the best possible result, namely:

Show that Minkowski's Theorem cannot be improved; that is, find an example of a centrally symmetric, convex subset $X$ of $V$ for which vol($X$) $=$ $2^{n}$vol($\Gamma$), and $X$ does not contain any nonzero lattice point in $\Gamma$.

I am unsure where to even begin looking for such a centrally symmetric, convex subset and even a complete lattice that would satisfy this property. The $2^n$ seems to imply that I'm missing something obvious, and I'm not generally adept at these more visual/geometric ideas. Would appreciate if someone could give a small indication.

Answer 1

Hint: First think about the case of $n=1$. Here the lattices are $\Gamma=\lambda\mathbb Z$ inside $\mathbb R$ with $\lambda>0$, and $\operatorname{Vol}(\Gamma)=\lambda$. The only convex (in fact, the only connected) subsets of $\mathbb R$ are intervals. What's the largest interval in $\mathbb R$ around $0$ which doesn't contain any nonzero element of $\lambda\mathbb Z$?

Now, think about $n=2$, and specialize to the case of $\Gamma=\mathbb Z^2$. Is there a way to generalize your construction from the $n=1$ case to a subset $X\subset\mathbb R^2$ with area $4$ but no nonzero lattice point? How does this generalize to other lattices?

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For the purposes of completeness, I'm putting a full-ish answer here, but in spoiler tags.

For $n=1$,

the largest interval without any nonzero element of $\lambda\mathbb Z$ is $(-\lambda,\lambda)$, which has volume $2\lambda=2\operatorname{Vol}(\Gamma)$.

For $n=2$ and $\Gamma=\mathbb Z^2\subset \mathbb R^2$,

you can set $X=(-\lambda,\lambda)^2$. In general, if $\Gamma=\mathbb Ze_1+\mathbb Ze_2$, you can set $X=(-1,1)e_1+(-1,1)e_2$; since every point in $\mathbb R^2$ is a unique linear combination of $e_1$ and $e_2$, this will never intersect a nonzero lattice point.

For general $n$ and $\Gamma$,

if $\Gamma$ is generated by $e_1,\dots,e_n$, set $$X=\sum_{i=1}^n (-1,1)e_i.$$ This works for the same reason as the $n=2$ case in the previous paragraph, and has by definition volume $2^n$ times that of $\sum [0,1)e_i$, i.e. $2^n\operatorname{Vol}(\Gamma)$.