# can minimal volume rational subspaces in a lattice be arbitrarily 'close'.

Let $\Gamma$ be a cocompact lattice in $\mathbb{R}^n$, eg. $\Gamma = A \mathbb{Z}^n$ for some $A \in SL_n \mathbb{R}$. Then any $k$-dimensional subspace $P$ which is rational in $\Gamma$ has a volume: ie. $P$ rational in $\Gamma$ means $P \cap \Gamma$ is a cocompact lattice in $P$ and we assign $vol(P, \Gamma)=covolume(P\cap \Gamma / P)$. Concretely if $x_1, \ldots, x_k$ is a $\mathbb{Z}$-basis for $P \cap \Gamma$, then $vol(P, \Gamma)^2=det(\langle x_i, x_j \rangle)$.

Now there is the following basic principle concerning minimal volume rational 1-dimensional subspaces of $\Gamma$ (ie. the shortest nonzero vectors of $\Gamma$, or $syst_1 (\Gamma)$): if $x, x'$ are distinct nonzero shortest lattice elements in $\Gamma$, then the angle between $x,x'$ is 'large'. This is just a computation: for if the angle between $x,x'$ is sufficiently small, then their difference $x-x'$ will be a shortER lattice element.

My question: is there an analogous principle for minimal volume rational $k$-dimensional subspaces in $\Gamma$? More specifically, suppose we consider the $\mathbb{R}$-span $W$ of the minimal volume $k$-dimensional subspaces in $\Gamma$. If $W \neq \mathbb{R}^n$, then there will exist a minimal volume rational $k$-plane $P$ which is not contained in $W$. If we had $k=1$, then we could say that the projection of $P$ onto the orthogonal complement of $W$ is 'large' (ie. $P$ has definite angle with $W$). What is the proper analogous statement for arbitrary $k$?

Remark (i): the principle for $k=1$ does not immediately extend to the cocompact lattice $\wedge^k \Gamma$ in $\wedge^k \mathbb{R}^n$, where indeed the minimal rational $k$-planes are just the shortest lattice elements in $\wedge^g \Gamma$. This because the difference $\xi-\xi'$ of two simple elements $\xi, \xi'$ does not itself have to be simple (ie. does not have to correspond to a $k$-plane).

Remark (ii): if we take an orthocomplement $V$ to $W$ in $\mathbb{R}^{n}$, then we have an orthogonal decomposition (wrt the induced inner product on $\wedge^k$): $$\wedge^k (W \oplus V)=\oplus_{a+b=k} \wedge^a W \otimes \wedge ^b V.$$

We can then also compute the norm of any rational subspace $Q$ as the sum of squares of its projections (seeing $Q$ as corresponding to a simple element in $\wedge^k$). Then maybe the best possible analogue is only the 'weak' statement that the projection onto $\wedge^k W \otimes \wedge^0 V$ is 'small'. But this seems almost technically useless. The point is that the principle for $k=1$ has some strength---we should then like to find something comparably useful for arbitrary $k$.

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