  Let $L$ be a lattice of full rank in $n$-dimensional real space. A vector in
$L$ is called $i$-sparse if it has no more than $i$ nonzero coordinates. We
define the $i$-th successive sparsity level of $L$, $s_i(L)$, to be the minimal
$s$ so that $L$ has $s$ linearly independent $i$-sparse vectors, then $s_i(L)
\leq n$ for each $1 \leq i \leq n$. We investigate sufficient conditions for
$s_i(L)$ to be smaller than $n$ and obtain explicit bounds on the sup-norms of
the corresponding linearly independent sparse vectors in~$L$. This result can
be viewed as a partial sparse analogue of Minkowski's successive minima
theorem. We then use this result to study virtually rectangular lattices,
establishing conditions for the lattice to be virtually rectangular and
determining the index of a rectangular sublattice. We further investigate the
$2$-dimensional situation, showing that virtually rectangular lattices in the
plane correspond to elliptic curves isogenous to those with real $j$-invariant.
We also identify planar virtually rectangular lattices in terms of a natural
rationality condition of the geodesics on the modular curve carrying the
corresponding points.
