A theorem on continued fractions

Duke’s Theorem and Continued Fractions

For uniformly chosen random α ∈ [0, 1], it is known the probability the nth digit of the continued-fraction expansion, [α]n converges to the Gauss-Kuzmin distribution P([α]n = k) ≈ log2(1 + 1/k(k + 2)) as n → ∞. In this paper, we show the continued fraction digits of √ d, which are eventually periodic, also converge to the Gauss-Kuzmin distribution as d → ∞ with bounded class number, h(d). The ...

Episturmian morphisms and a Galois theorem on continued fractions

We associate with a word w on a finite alphabet A an episturmian (or Arnoux-Rauzy) morphism and a palindrome. We study their relations with the similar ones for the reversal of w. Then when |A| = 2 we deduce, using the Sturmian words that are the fixed points of the two morphisms, a proof of a Galois theorem on purely periodic continued fractions whose periods are the reversal of each other. Ma...

On multidimensional generalization of the Lagrange theorem on continued fractions ∗

We prove a multidimensional analogue of the classical Lagrange theorem on continued fractions. As a multidimensional generalization of continued fractions we use Klein polyhedra.

CMFT-MS 15094 The parabola theorem on continued fractions

Using geometric methods borrowed from the theory of Kleinian groups, we interpret the parabola theorem on continued fractions in terms of sequences of Möbius transformations. This geometric approach allows us to relate the Stern–Stolz series, which features in the parabola theorem, to the dynamics of certain sequences of Möbius transformations acting on three-dimensional hyperbolic space. We al...

On a theorem of Legendre in the theory of continued fractions