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On the Quantitative Metric Theory of Continued Fractions in Positive Characteristic
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 61 / Issue 1 / February 2018
- Published online by Cambridge University Press:
- 07 February 2018, pp. 283-293
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- Article
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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
where
$$\alpha = A_0(\alpha ) + \displaystyle{1 \over {A_1(\alpha ) + \displaystyle{1 \over {A_2(\alpha ) + \ddots }}}},$$
$(A_{n}(\alpha))_{n=0}^{\infty}$ 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
for almost everywhere α with respect to Haar measure.
$$\vert A_1(\alpha ) \ldots A_N(\alpha )\vert ^{1/N} = q^{q/(q - 1)} + o(N^{ - 1/2}(\log N)^{3/2 + {\rm \epsilon }})$$
