Simple continued fractions for some irrational numbers. (English) Zbl 0404.10003

Shallit, Jeffrey

J. Number Theory 11, 209-217 (1979).

Much is known about the continued fraction expansion of a real number \(x\) and the connection between this expansion and algebraic properties of \(x\) [see for instance: O. Perron, Die Lehre von den Kettenbrüchen. 2. Aufl. Leipzig: B. G. Teubner (1929; JFM 55.0262.09)]. Except for terminating and periodic expansions (rational \(x\) and algebraic \(x\) of degree 2 resp.) there are more well known patterns: in the expansion for \(e, e^2, \tanh(1/h), \exp(1/m), \exp(2/m)\) etc.
In the paper under review the author derives definite information on the pattern in the expansion of the transcendental numbers \(B(u,\infty) = \sum_{k=0}^\infty u^{-2^k}\) \((u\ge 3\), an integer). The proofs are elegant and only use well known properties of continued fractions and the principle of induction. Using \(B(u,v)\) for the partial sums \(\sum_{k=0}^v\) and \([a_0, a_1, a_2,\ldots]\) for the continued fraction \(a_0 + \frac{1 \vert}{\vert a_1} + \frac{1 \vert}{\vert a_2} + \cdots\), the results are: \[ B(u,0) = [0,u],\quad B(u,1) = [0,u -1,u +1], \quad B(u,v)=[a_0, a_1, \ldots a_n] \tag{1} \Rightarrow \] 2. \(\quad\) the expansion for \(B(u,v)\) has \(2^v+1\) partial denominators of which the first \(2^v\) are identical to those in the expansion for \(B(u,\infty)\).
\[ B(u,\infty) = [a_0, a_1, a_2,\ldots] \Rightarrow B(u+b,\infty) = [a_0, a_1+b, a_2+b,\ldots] \quad (u\ge 3,\ b\ge 0). \tag{3} \] 4. The mass (M. Shrader-Frechette) of \(B(u,v)\) is \(u\cdot 2^v\).
5. \(B(u,\infty)\) has five unique partial denominators (thus bounded): \(0, u-2, u-1, u\) and \(u+2\). The number of occurrences of these numbers in the expansion for \(B(u,v)\) is \(1, 2^{v2}- 1, 2, 2^{v-1}-1\) and \(2^{v-2}\), respectively.
6. If \(B(u,\infty) = [a_0, a_1,\ldots]\) then \(a_n = u+2\) if \(n\equiv 2\text{ or }7 \bmod 8\) and \(= u\) if \(n\equiv 3\text{ or }6 \bmod 8\). Furthermore the numbers arising from adding a factor \((-1)^k\) in \(B(u,\infty)\) are considered.

Reviewer: Marcel G. de Bruin (Amsterdam)

MSC:

11A55	Continued fractions
11J70	Continued fractions and generalizations

Keywords:

continued fraction expansion; pattern; transcendental numbers

Citations:

JFM 55.0262.09

References:

[1]	Perron, O., (Die Lehre von den Kettenbrüchen, Vol. 1 (1957), Teubner: Teubner Stuttgart) · JFM 55.0262.09
[2]	Hardy, G. H.; Wright, E. M., (An Introduction to the Theory of Numbers (1971), Clarendon: Clarendon Oxford)
[3]	Shrader-Frechette, M., (Continued Fractions, Farey Fractions, and Pascal’s Triangle. Continued Fractions, Farey Fractions, and Pascal’s Triangle, presented at the Miami University Number Theory Conference (September 30, 1977))
[4]	Schneider, T., (Einführung in die Transzendenten Zahlen (1957), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0077.04703
[5]	Khintchine, A. Ya, (Continued Fractions (1963), P. Noordhoff: P. Noordhoff Groningen) · Zbl 0117.28503