On the Algebraic Independence Of P−Adic Continued Fractions

On the complexity of algebraic numbers, II. Continued fractions

Let b 2 be an integer. Émile Borel [9] conjectured that every real irrational algebraic number α should satisfy some of the laws shared by almost all real numbers with respect to their b-adic expansions. Despite some recent progress [1], [3], [7], we are still very far away from establishing such a strong result. In the present work, we are concerned with a similar question, where the b-adic ex...

Algebraic continued fractions in F q ( ( T −

On the entropy of Japanese continued fractions

We consider a one-parameter family of expanding interval maps {Tα}α∈[0,1] (japanese continued fractions) which include the Gauss map (α = 1) and the nearest integer and by-excess continued fraction maps (α = 1 2 , α = 0). We prove that the Kolmogorov-Sinai entropy h(α) of these maps depends continuously on the parameter and that h(α) → 0 as α → 0. Numerical results suggest that this convergence...

On the Extremal Theory of Continued Fractions

Letting x = [a1(x), a2(x), . . .] denote the continued fraction expansion of an irrational number x ∈ (0, 1), Khinchin proved that Sn(x) = ∑n k=1 ak(x) ∼ 1 log 2 n logn in measure, but not for almost every x. Diamond and Vaaler showed that removing the largest term from Sn(x), the previous asymptotics will hold almost everywhere, showing the crucial influence of the extreme terms of Sn(x) on th...

On the Maillet–baker Continued Fractions

We use the Schmidt Subspace Theorem to establish the transcendence of a class of quasi-periodic continued fractions. This improves earlier works of Maillet and of A. Baker. We also improve an old result of Davenport and Roth on the rate of increase of the denominators of the convergents to any real algebraic number.