Continued Fractions With Bounded Partial Quotients

Abstract

If α is not a quadratic irrational, then we produce a specific sequence of quadratic irrational approximations to α, the rate of convergence given in terms of L and γ. As an application, we demonstrate the transcendence of some continued fractions, a typical one being of the form [0, u 1 , u 2 , . . . ] with um = 1 + mθ mod n, n 2, and θ an irrational number which satisfies any of a given set of conditions.

On the complexity of algebraic numbers, II. Continued fractions

“…Apart from this assumption, Theorem 1 does not depend on the size of the partial quotients of α. This is in a striking contrast to all previous results [5,13,8], in which, roughly speaking, the size w of the repetition is required to be all the more large than the partial quotients are big. Unlike these results, our Theorem 1 can be easily applied even if α has unbounded partial quotients.…”

“…It may denote real numbers whose continued fraction expansion has some regularity, or can be produced by a simple algorithm (by a simple Turing machine, for example), or arises from a simple dynamical system... The main results of the present work are two new combinatorial transcendence criteria, which considerably improve upon those from [5,13,8]. It is of a particular interest that such criteria naturally yield, in a unified way, several new results on the different approaches of the above mentioned notion of simplicity/complexity for the continued fraction expansions of algebraic real numbers of degree at least three.…”

Maximal continuants and the Fine–Wilf theorem

“…This beautiful result reveals a new aspect of the interplay between analysis and combinatorical algebra. In fact, Baxa's method appears to be as powerful as a different approach to the same question via a semigroup of matrices due to Allouche, Davison, Queffélec and Zamboni (see [1,5]). …”

Quasi-periodic continued fractions

“…Such techniques have occurred sporadically in recent years with different notation and emphasis. The reader can consult [6][7][8]10] for some related work.…”