关于稀疏数的几何
On sparse geometry of numbers
摘要：
令$L$是$n$维实空间中的满秩格。$L$中的向量称为$i$ -稀疏，如果它不超过$i$非零坐标。我们定义$L$的第$i$个连续稀疏水平$s_i(L)$为最小$s$，使得$L$具有$s$线性独立的$i$ -稀疏向量，则每个$1 \leq i \leq n$的$s_i(L) \leq n$。我们研究了$s_i(L)$小于$n$的充分条件，并得到了$L$中相应线性无关稀疏向量的次范数的显式界.这个结果可以看作是Minkowski连续极小定理的部分稀疏模拟。然后，我们使用这个结果来研究虚拟矩形格，建立条件的晶格是虚拟矩形和确定的矩形子格的指数。我们进一步调查的$2$维的情况下，显示在平面上的几乎矩形晶格对应于椭圆曲线isogenous那些真正的$j$不变。我们还确定平面虚拟矩形格的自然合理性条件的测地线上的模块化曲线进行相应的点。
英文摘要：
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.
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