Continued fractions with bounded partial quotients

Precise bounds are given for the quantity [GRAPHICS] where (q(m)) is the classical sequence of denominators of convergents to the continued fraction alpha = [0, u(1), u(2),...] and (u(m)) is assumed bounded, with a distribution. If the infinite word u = u(1)u(2) ... has arbitrarily large instances of segment repetition at or near the beginning of the word, then we quantify this property by means of a number gamma, called the segment-repetition factor. If alpha is not a quadratic irrational, then we produce a specific sequence of quadratic irrational approximations to alpha, the rate of convergence given in terms of L and gamma. As an application, we demonstrate the transcendence of some continued fractions, a typical one being of the form [0, u(1), u(2),....] with u(m) = 1 + left perpendicular mtheta right perpendicular mod n, n greater than or equal to 2, and theta an irrational number which satisfies any of a given set of conditions.