Let pₙ/qₙ = [a₀;a₁,...,aₙ] be the n-th convergent of the continued fraction expansion of [a₀;a₁,a₂,...]. Leaping convergents are those of every r-th convergent $p_{rn+i}/q_{rn+i}$ (n = 0,1,2,...) for fixed integers r and i with r ≥ 2 and i = 0,1,...,r-1. The leaping convergents for the e-type Hurwitz continued fractions have been studied. In special, recurrence relations and explicit forms of such leaping convergents have been treated. In this paper, we consider recurrence relations and explicit forms of the leaping...

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 $p_{rn+i}/q_{rn+i}$ (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 $l\left({\left(\surd d\right)}^{2n+1}\right)$ be the length of the continued fraction expansion of ${\left(\surd d\right)}^{2n+1}$. 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 $C₁{\left(\surd d\right)}^{2n+1}\ge l\left({\left(\surd d\right)}^{2n+1}\right)\ge C₂{\left(\surd d\right)}^{2n+1}$ 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].

We consider the ensemble of curves {γα, N: α∈(0, 1], N∈ℕ} obtained by linearly interpolating the values of the normalized theta sum N−1/2∑n=0N'−1exp(πin2α), 0≤N'<N. We prove the existence of limiting finite-dimensional distributions for such curves as N→∞, when α is distributed according to any probability measure λ, absolutely continuous w.r.t. the Lebesgue measure on [0, 1]. Our Main Theorem generalizes a result by Marklof [Duke Math. J.97 (1999) 127–153] and Jurkat and van Horne [Duke...

Let $\theta$ be a real number with continued fraction expansion$$\theta =\left[a_0,a_1,a_2,\cdots \right],$$and let$$M=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$$be a matrix with integer entries and nonzero determinant. If $\theta$ has bounded partial quotients, then $\frac{a\theta +b}{c\theta +d}=\left[a_0^{*},a_1^{*},a_2^{*},\cdots \right]$ also has bounded partial quotients. More precisely, if $a_j\le K$ for all sufficiently large $j$, then $a_j^{*}\le |det\left(M\right)|(K+2)$ for all sufficiently large $j$. We also give a weaker bound valid for all $a_j^{*}$ with $j\ge 1$. The proofs use the homogeneous Diophantine approximation constant $L_{\infty }\left(\theta \right)={lim\; sup}_{q\to \infty }{\left(q∥q^{\theta }∥\right)}^{-1}$. We show that$$\frac{1}{\left|det\left(M\right)\right|}{L}_{\infty }\left(\theta \right)\le L_{\infty }\left(\frac{a\theta +b}{c\theta +d}\right)\le \left|det\left(M\right)\right|L_{\infty }\left(\theta \right).$$

The main result of this paper is a criterion for linear independence of continued fractions over the rational numbers. The proof is based on their special properties.