On the metric theory of nearest integer continued fractions (Q1821813)

The first part of this paper is a short introduction to some results, such as P. Lévy's and Khintchine's, in the metrical theory of continued fraction expansions. References are to Khintchine's well-known book from 1935, an excellent introduction in its days, but since the work of W. Doeblin (1940) and C. Ryll-Nardzewski (1951) who revealed the ergodic theoretic background of these results, somewhat out-dated. The present paper ignores this important aspect of the theory. In the second part the author tries to give analogous results for the nearest integer continued fraction. He asks in fact for the measure which, together with the one-sided shift for the nearest integer continued fraction, yields an ergodic system. A certain function, called m, plays a crucial rôle. This m is given as the limit of a sequence of functions \(m_ n\) which are obtained recursively. Tables for m and m' are provided and values for certain limits derived from it. Apparently the author is unaware of the fact that the questions he raises have already been completely answered by \textit{G. J. Rieger} [Ein Gauss- Kusmin-Lévy-Satz für Kettenbrüche nach nächsten Ganzen, Manuscr. Math. 24, 437-448 (1978; Zbl 0377.10028) and Mischung und Ergodizität bei Kettenbrüchen nach nächsten Ganzen, J. Reine Angew. Math. 310, 171-181 (1979; Zbl 0409.10038)], by \textit{A. M. Rockett} [The metrical theory of continued fractions to the nearer integer, Acta Arith. 38, 97- 103 (1980; Zbl 0368.10039)], not to mention \textit{H. Nakada} [Metrical theory for a class of continued fraction transformations and their natural extensions, Tokyo J. Math. 4, 399-426 (1981; Zbl 0479.10029)] who gives the natural extension of the ergodic system in question. For the latest metrical results for this continued fraction one should consult a recent paper by \textit{C. Kraaikamp} [The distribution of some sequences connected with the nearest integer continued fraction, Indagationes Math. 49, 177-191 (1987)]. From Rieger and Rockett we see that the function m' from the paper is in fact the function \(\frac{1}{\log G}(\frac{1}{G+t}+\frac{1}{G+1-t}),\) \(G:=(\sqrt{5}+1)/2.\) Checking the tables for m' with this expression one finds that all the last decimals (of the five) are incorrect. How errors propagate is shown by the claim that \(B_ n^{1/n}\approx 5.4564...\), formula (42), whereas in fact this constant is \(\exp (\pi^ 2/12 \log G)=5.524307...\) It is quite understandable that a numerical analyst is not acquainted with recent results in number theory but the redaction of the Journal has failed in its important task to spare author and readers the publication of a superfluous paper. \{Editor's remark: The editor of BIT will publish a short note in the March 1988 issue on this paper.\}