Minimal vector systems in three-dimensional lattices and an analogue of Vahlen’s theorem for three-dimensional Minkowski continued fractions

Abstract

There exist two geometric interpretations of classical continued fractions admitting a natural generalization to the multidimensional case. In one of these interpretations, which is due to Klein (see [1, 2]), a continued fraction is identified with the convex hull (the Klein polygon) of the set of integer lattice points belonging to two adjacent angles. The second interpretation, which was independently proposed by Voronoi and Minkowski (see [3–6]), is based on local minima of lattices, minimal systems, and extremal parallelepipeds (the definitions of the notions mentioned in the introduction are given later on). The vertices of Klein polygons in plane lattices can be identified with local minima; however, beginning with the dimension 3, the Klein and Voronoi–Minkowski geometric constructions become different (see [7, 8]). The constructions of Voronoi and Minkowski is simpler from the computational point of view. In particular, they make it possible to design efficient algorithms for determining fundamental units in cubic fields. In both Voronoi’s and Minkowski’s approaches, the three-dimensional theory of continued fractions is based on beautiful theorems of the geometry of numbers (a discussion and a reexposition of the original results are contained in monographs [9–11]). Moreover, both of them use the natural assumption that the lattices under consideration are irreducible (that is, none of the coordinates of any two different lattice points coincide). In particular, lattices of cubic irrationalities are irreducible. However, some number-theoretic problems, such as those related to analyzing properties of Korobov grids (see [12–14]), require studying local properties of integer lattices, which are not irreducible. In [15], it was mentioned that Vahlen’s theorem on the approximation of numbers by convergents (see [16, 17]) in terms of lattices admits the following interpretation: If vectors γa = (a1, a2) and γb = (−b1, b2) form a Voronoi basis, then min{a1a2, b1b2} ≤ 1 2 det Γ.