Lattice distances in 3-dimensional quantum jumps
Let \(P \subset \mathbb{R}^d\) be a \textit{lattice polytope}, i.e., the convex hull of finitely many points in \(\mathbb{Z}^d\). If \(x \in\mathbb{Z}^d \setminus P\) then we say \((P, x)\) is a \textit{quantum jump} if \[ \mathrm{conv} (P \cup \{ x \}) \cap\mathbb{Z}^d \ = \ \left( P \cap\mathbb{Z}^d \right) \cup \{ x \}. \] An affine functional \(f:\mathbb{R}^d \to\mathbb{R}\) is \textit{primitive} if \(f(\mathbb{Z}^d) = \mathbb{Z}\). The \textit{lattice distance} between a rational hyperplane \(H \subset\mathbb{R}^d\) and a lattice point \(x \in\mathbb{Z}^d\) is \(\mathrm{dist}(x, H) := |f(x)|\) where \(f\) is a primitive functional with \(f(H) = 0\). If \(P\) is a \((d-1)\)-dimensional lattice polytope such that \(x \in\mathbb{Z}^d\) is not in the affine hull \(\mathrm{aff}(P)\) of \(P\), the lattice distance function naturally extends to \(\mathrm{dist}(x, P) :=\mathrm{dist}(x, \mathrm{aff}(P))\). This notion further extends to the lattice distance between \(x\) and a full-dimensional polytope \(P\) via a visible facet construction. The paper under review collects several results on bounds for the lattice distance in 3-dimensional quantum jumps. To give one example, the lattice distance between \(x \in\mathbb{Z}^3 \setminus P\) and a lattice polygon \(P \subset\mathbb{R}^3\) is at most 3, unless \(P\) is a lattice triangle of width 1, in which case the distance is unbounded. For the entire collection see [Zbl 1497.52003].
scientific article; zbMATH DE number 7632921