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 ith successive sparsity level of L, \(s_i(L)\), to be the minimal s so that L has s linearly independent i-sparse vectors, then \(s_i(L) \le n\) for each \(1 \le i \le n\). 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. These results 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.

令L为n维实空间中满秩的晶格。如果L中的向量不超过i个非零坐标，则称为i-稀疏。我们定义我的个连续的稀疏水平大号，\（S_I（L）\） ，是最小的小号，使得大号具有š线性独立我-sparse载体，然后\（S_I（L）\了N \）为每个\（1 \ le i \ le n \）。我们调查足以使\（s_i（L）\）小于n的条件并获得L中对应的线性独立稀疏向量的sup-范数的显式边界。这些结果可以看作是Minkowski连续极小定理的部分稀疏类似物。然后，我们使用此结果来研究虚拟的矩形晶格，为晶格变为虚拟的矩形建立条件，并确定矩形子晶格的索引。我们进一步研究二维情况，表明平面中的几乎矩形格子对应于椭圆曲线，而这些椭圆曲线与具有真正j不变的椭圆曲线是同质的。我们还根据载有相应点的模块化曲线上测地线的自然合理性条件，确定了平面虚拟的矩形格子。