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On the Quantitative Metric Theory of Continued Fractions in Positive Characteristic

Part of: Measure-theoretic ergodic theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  07 February 2018

Poj Lertchoosakul  and
Radhakrishnan Nair
Poj Lertchoosakul
Affiliation:
Instytut Matematyki, Uniwersytet Gdanski, ul. Wita Stwosza 57, 80-308 Gdansk, Poland ([email protected])
Radhakrishnan Nair*
Affiliation:
Mathematical Sciences, the University of Liverpool, Peach Street, Liverpool L69 7ZL, UK ([email protected])
*
*Corresponding author.
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Abstract

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Let 𝔽q be the finite field of q elements. An analogue of the regular continued fraction expansion for an element α in the field of formal Laurent series over 𝔽q is given uniquely by

$$\alpha = A_0(\alpha ) + \displaystyle{1 \over {A_1(\alpha ) + \displaystyle{1 \over {A_2(\alpha ) + \ddots }}}},$$
where $(A_{n}(\alpha))_{n=0}^{\infty}$ is a sequence of polynomials with coefficients in 𝔽q such that deg(An(α)) ⩾ 1 for all n ⩾ 1. In this paper, we provide quantitative versions of metrical results regarding averages of partial quotients. A sample result we prove is that, given any ϵ > 0, we have
$$\vert A_1(\alpha ) \ldots A_N(\alpha )\vert ^{1/N} = q^{q/(q - 1)} + o(N^{ - 1/2}(\log N)^{3/2 + {\rm \epsilon }})$$
 for almost everywhere α with respect to Haar measure.

Keywords

continued fractionsmetric theory of numbers

MSC classification

Primary: 11K50: Metric theory of continued fractions
Secondary: 28D99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 1 , February 2018 , pp. 283 - 293
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Edinburgh Mathematical Society 2018

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