论文标题
计数晶格中的基础扩展
Counting basis extensions in a lattice
论文作者
论文摘要
鉴于整数晶格中的原始矢量集合，我们计算了可以将其扩展到基于$ t $的向量将其扩展到基础的方式，从而产生了渐近估计值为$ t \ f \ infty $。可以用单模型矩阵以及一类多线性形式的表示问题来解释此问题。在$ 2 $维的情况下，此问题也与Farey分数的分布有关。作为辅助引理，我们证明了对〜$ \ Mathbb r^n $中的超平面中有限sup-norm的整数晶格点的计数估算值。我们对计数基础扩展的主要结果也将〜$ \ mathbb r^n $中的任意晶格推广到任意晶格。最后，我们通过多线性形式建立了整数稀疏表示的一些基本属性。
Given a primitive collection of vectors in the integer lattice, we count the number of ways it can be extended to a basis by vectors with sup-norm bounded by $T$, producing an asymptotic estimate as $T \to \infty$. This problem can be interpreted in terms of unimodular matrices, as well as a representation problem for a class of multilinear forms. In the $2$-dimensional case, this problem is also connected to the distribution of Farey fractions. As an auxiliary lemma we prove a counting estimate for the number of integer lattice points of bounded sup-norm in a hyperplane in~$\mathbb R^n$. Our main result on counting basis extensions also generalizes to arbitrary lattices in~$\mathbb R^n$. Finally, we establish some basic properties of sparse representations of integers by multilinear forms.
