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We establish the (level-1) large deviation principles for three kinds of means associated with the backward continued fraction expansion. We show that: for the harmonic and geometric means, the rate functions vanish exactly at one point; for the arithmetic mean, it is completely degenerate, vanishing at every point in its effective domain. Our method of proof employs the thermodynamic formalism for finite Markov shifts, and a multifractal analysis for the Renyi map generating the backward continued fraction digits. We completely determine the class of unbounded arithmetic functions for which the rate functions vanish at every point in unbounded intervals.
We show that the growth rate of denominator $Q_n$ of the $n$-th convergent of negative expansion of $x$ and the rate of approximation: $$ frac{log{n}}{n}log{left|x-frac{P_n}{Q_n}right|}rightarrow -frac{pi^2}{3} quad text{in measure.} $$ for a.e. $x$.
Large and moderate deviation principles are proved for Engel continued fractions, a new type of continued fraction expansion with non-decreasing partial quotients in number theory.
In this paper, we represent a continued fraction expression of Mathieu series by a continued fraction formula of Ramanujan. As application, we obtain some new bounds for Mathieu series.
We present a large deviation principle for the entropy penalized Mather problem when the Lagrangian L is generic (in this case the Mather measure $mu$ is unique and the support of $mu$ is the Aubry set). Consider, for each value of $epsilon $ and h,
We adjust Arnouxs coding, in terms of regular continued fractions, of the geodesic flow on the modular surface to give a cross section on which the return map is a double cover of the natural extension for the alpha-continued fractions, for each $alp