论文标题
在数字的稀疏几何形状上
On sparse geometry of numbers
论文作者
论文摘要
让$ l $成为$ n $维真实空间的全等级。 $ l $的矢量如果不超过$ i $ nonzero坐标，则称为$ i $ -sparse。我们定义了$ i $ th的连续稀疏级别的$ l $，$ s_i（l）$，为最小$ s $，以便$ l $具有$ s $ s $ s $ linearly Independent $ i $ i $ i $ -sparse vectors，然后是$ s_i（l）\ leq n $，每个$ 1 \ leq leq i \ leq i \ leq n $。我们调查了$ s_i（l）$的足够条件，使其小于$ n $，并在〜$ l $中的相应线性独立稀疏向量的Sup-Norms上获得明确的界限。该结果可以看作是Minkowski连续的Minima定理的部分稀疏类似物。然后，我们使用此结果来研究几乎矩形的晶格，从而确定晶格实际上是矩形并确定矩形sublattice的索引。我们进一步调查了$ 2 $维的情况，表明飞机上的矩形晶格与真正的$ j $ invariant的椭圆曲线相对应。我们还根据携带相应点的模块化曲线上的大地曲线的自然合理性条件来识别平面矩形晶格。
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.
