Denote the n-th convergent of the continued fraction by pₙ/qₙ = [a₀;a₁,...,aₙ]. We give some explicit forms of leaping convergents of Tasoev continued fractions. For instance, [0;ua,ua²,ua³,...] is one of the typical types of Tasoev continued fractions. Leaping convergents are of the form (n=0,1,2,...) for fixed integers r ≥ 2 and 0 ≤ i ≤ r-1.
Let d ≥ 2 be a square-free integer and for all n ≥ 0, let be the length of the continued fraction expansion of . If ℚ(√d) is a principal quadratic field, then under a condition on the fundamental unit of ℤ[√d] we prove that there exist constants C₁ and C₂ such that for all large n. This is a generalization of a theorem of S. Chowla and S. S. Pillai [2] and an improvement in a particular case of a theorem of [6].
In this paper we describe the set of conjugacy classes in the group . We expand geometric Gauss Reduction Theory that solves the problem for to the multidimensional case, where -reduced Hessenberg matrices play the role of reduced matrices. Further we find complete invariants of conjugacy classes in in terms of multidimensional Klein-Voronoi continued fractions.