Geometry and combinatoric of Minkowski-Voronoi 3-dimensional continued fractions (Q518096)

It is well known that if \((a,N)\), \(a< N,\) is a pair of positive integers then the lengths of ordinary continued fractions for \[ \frac{N}{a}, \; \frac{N+a}{a},\; \frac{N+2a}{a},\; \frac{N+3a}{a}, \dots \] coincide. The authors generalize this statement to the case of triples \((a, b, N)\) where \(a\) and \(b\) are relatively prime with \(N\). They also discuss the Minkowski-Voronoi generalization of the continued fractions defined by such triples. It was showed that the space of all triples (with additional constrains: \(b \geq 2\) and \(N\) is not divisible by \(b\)) splits into \(2\)-dimensional families such that the Minkowski-Voronoi continued fractions in each of these families are almost all combinatorially equivalent to each other. The asymptotic stability of Minkowski-Voronoi complexes in special two-parametric families of rank-1 lattices is proved. The complexes for the case of White's rank-1 lattices are constructed explicitly and provided with a hypothetic description in more complicated settings.