One important question in the theory of lattices is to detect a shortest vector: given a norm and a lattice, what is the smallest norm attained by a non-zero vector contained in the lattice? We focus on the infinity norm and work with lattices of the form $A\mathbb{Z}^n$, where $A$ has integer entries and is of full column rank. Finding a shortest vector is NP-hard. We show that this task is fixed parameter tractable in the parameter $\Delta$, the largest absolute value of the determinant of a full rank submatrix of $A$. The algorithm is based on a structural result that can be interpreted as a threshold phenomenon: whenever the dimension $n$ exceeds a certain value determined only by $\Delta$, then a shortest lattice vector attains an infinity norm value of one. This threshold phenomenon has several applications. In particular, it reveals that integer optimal solutions lie on faces of the given polyhedron whose dimensions are bounded only in terms of $\Delta$.

翻译：格理论中的一个重要问题是检测最短向量：给定范数和格，格中包含的非零向量能达到的最小范数是多少？我们聚焦于无穷范数，研究形式为$A\mathbb{Z}^n$的格，其中$A$具有整数条目且列满秩。寻找最短向量是NP难问题。我们证明该任务在参数$\Delta$（$A$的满秩子矩阵行列式绝对值的最大值）下是固定参数可处理的。该算法基于一个可解释为阈值现象的结构性结果：当维度$n$超过仅由$\Delta$确定的某个值时，最短格向量将达到无穷范数值一。这一阈值现象具有若干应用。特别地，它揭示了整数最优解位于给定多面体的面上，而这些面的维度仅由$\Delta$界定。