A lattice \(\Lambda \) is said to be an extension of a sublattice L of smaller rank if L is equal to the intersection of \(\Lambda \) with the subspace spanned by L. The goal of this paper is to initiate a systematic study of the geometry of lattice extensions. We start by proving the existence of a small-determinant extension of a given lattice, and then look at successive minima and covering radius. To this end, we investigate extensions (within an ambient lattice) preserving the successive minima of the given lattice, as well as extensions preserving the covering radius. We also exhibit some interesting arithmetic properties of deep holes of planar lattices.

如果L等于\(\Lambda \)与L所跨越的子空间的交集，则称格\(\Lambda \)为较小秩的子格L的扩展。本文的目标是启动对晶格延伸几何的系统研究。我们首先证明给定格的小行列式扩展的存在性，然后研究连续的最小值和覆盖半径。为此，我们研究了保留给定晶格的连续最小值的扩展（在环境晶格内），以及保留覆盖半径的扩展。我们还展示了平面晶格深孔的一些有趣的算术性质。