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Linear relations among asymptotic frequencies in continued fractions

  • Kurt GirstmairEmail author
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Abstract

Let H(md) denote the asymptotic frequency of the natural numbers \(k\equiv d \mod m\) in the continued fraction expansions of almost all numbers \(x\in [0,1)\). For a fixed number \(m\ge 4\), we study \(\mathbb {Q}\)-linear relations among the numbers H(md), \(1\le d\le m-3\), i.e., vectors \((c_1,\ldots ,c_{m-3})\in \mathbb {Q}^{m-3}\) such that
$$\begin{aligned} \sum _{d=1}^{m-3} c_dH(m,d)=0. \end{aligned}$$
We restrict ourselves to the symmetric case \(c_d=c_{m-2-d}\). In the end, we obtain a basis of the \(\mathbb {Q}\)-vector space of these relations for prime powers m and for \(m=pq\), where \(p\ne q\) are primes.

Keywords

Asymptotic frequencies in continued fractions Linear relations among transcendental numbers Cyclotomic units 

Mathematics Subject Classification

11K50 11J81 

Notes

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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für MathematikUniversität InnsbruckInnsbruckAustria

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