ACTA issues

On calculating a continued fraction expansion from a decimal expansion

C. Faivre

Acta Sci. Math. (Szeged) 67:3-4(2001), 505-519

Abstract. For an irrational number $x$, let $k_n(x)$ be the number of partial quotients in the continued fraction expansion given by the first $n$-th decimals of $x$. A basic result of G. Lochs states that $$\lim_{n\to\infty }{k_n(x)\over n}={6\log2\log10\over\pi ^2}\approx0.9702$$ for almost all $x$ in the sense of Lebesgue. In this paper we give a survey of properties of $k_n$. New results are also proved. First we give a bound for the number of decimals to have a prescribed number of partial quotients. We prove also that if $x$ has a Lévy constant $\beta(x)$ then ${k_n(x)\over n}$ converges to the limit $\log10\over2\beta(x)$ if a condition on the growth of partial quotients is satisfied. This result improves the theorem of Lochs and has an important application to the case of quadratic numbers. We study also the possible positive limits of the sequence ${k_n(x)\over n}$ for all irrationals $x$. Finally a condition which ensures that ${k_n(x)\over n}$ converges to 0 is given. For example this condition holds for $x=e$, thus $e$ doesn't satisfy Loch's result.

AMS Subject Classification (1991): 11K50, 11Y65, 11A55

Keyword(s): Continued fraction expansion, decimal expansion, Lévy constants

Received July 6, 2000, and in revised form March 2, 2001. (Registered under 2800/2009.)