C327. Addenda for “the fractional dimensional theory of continued fractions”, proc. cambridge philos. soc. 37 (1941), 199-228.
C327. Addenda for “the fractional dimensional theory of continued fractions”, proc. cambridge philos. soc. 37 (1941), 199-228.
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DOI：
10.1080/00949658908811155
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发表时间：
1989-05
影响因子：
1.2
通讯作者：
I. Good
中科院分区：
文献类型：
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作者：
I. Good
在1953年和1954年，我与当时在美因茨大学数学研究所的维尔纳·弗里奇有过一些通信，他在信中指出了上述文章中的一些错误和含糊之处，我称之为G。我保存了这些通信，但丢失了我1940年的笔记。有些含糊其辞可能是由于需要压缩论点的结果，因为论文的篇幅已经超过了官方规定的最多20页印刷页。这篇论文被授予史密斯奖。由于现在人们对分形集及其(Hausdorff-Besicovitch)分维或分维(例如，Billingsley，1960；Falconer，1985；Feigbaum，1979；Mandelbrot，1982；Rogers，1970)以及由连分式定义的集合的分维理论(Beardon，1965；Bumby，1982,1985；Cusick，1977；Evans，1956；Hensley，1988a，b；Hirst，1973年；Rogers，1970，pp.135-147；Volkmann，1953154)似乎有很大的兴趣，因此似乎应该发表一些关于密切相关工作的额外评论。更正将发表在Good(1989)上，而本注解则提供补充注释。E，被定义为所有部分商都是1或2的简单连分式的集合。因此，它实际上包括除由J5生成的域中的数之外的所有“最差逼近”数(Hensley，1988b)，当然它与具有有界数字的连续数理论(Cusick，1977)有关。G表示0.5306<；dimE2~0.5320，其中dim E，表示E的分数维。Hensley(1988a)使用一种方法和一台个人计算机，将区间缩小到(0.53128049，0.53128051)。(因此，G(21.2)中不完全证明的上下界的几何平均值正确到小数点后五位。)Bumby(1973,1982)指出了马尔可夫谱理论的一个应用。虽然G关注的是连分式，但在导言中，人们猜想其数字具有极限频率p，，p1，…，p9的小数集具有分数维-xi pi log，，pi。埃格尔斯顿(1949)证明了这一点。关于这一结果的进一步简短评论，关于熵，见Good(1950b，第170页)和Shannon(1950，第174页)。这一结果在没有Good(1951)证明的情况下被推广到广义小数。关于广义小数，另见Everett(1946)。该结果被进一步推广到一阶马氏链上，并被Billingsley(1960)证明。他的结果可以用以下不那么抽象的方式重新表达(本着广义小数的精神)。设单位间隔被分解为
In 1953 and 1954 I had some correspondence with Werner Fritsch, then at the Mathematische Institut der Universitat Mainz, in which he pointed out some errors and obscurities in the abovementioned article which I shall call G. I have preserved the correspondence but have lost my notes of 1940. Some of the obscurities may have been a consequence of the need to compress the arguments since the paper was already longer than the official maximum of twenty printed pages. The paper was awarded a Smith's Prize. Because there is now much interest in fractal sets and in their (Hausdorff-Besicovitch) fractional dimensions or fractal dimensions (for example, Billingsley, 1960; Falconer, 1985; Feigenbaum, 1979; Mandelbrot, 1982; Rogers, 1970) and in the fractional dimensional theory of sets defined by continued fractions (Beardon, 1965; Bumby, 1982, 1985; Cusick, 1977; Evans, 1956; Hensley, 1988a, b; Hirst, 1970, 1973; Rogers, 1970, pp. 135-147; Volkmann, 1953154) it seems appropriate to publish the corrections, plus some additional comments concerning closely related work. The corrections are to be published in Good (1989) while the present note provides the additional comments.E, is defined as the set of simple continued fractions all of whose partial quotients are either 1 or 2. It thus consists in effect of all the" most badly approximable" numbers (Hensley, 1988b), apart from numbers in the field generated by J5, and is of course related to the theory of continuants with bounded digits (Cusick, 1977). G showed that 0.5306< dimE2~ 0.5320 where dim E, denotes the fractional dimension of E,. Hensley (1988a), by using a method, and a personal computer, narrowed the interval to (0.53128049, 0.53128051).(The geometric mean of the incompletely proved upper and lower bounds in G (21.2) is thus correct to five decimal places.) Bumby (1973, 1982) pointed out an application to the theory of Markov spectra. Although G was concerned with continued fractions, in the Introduction it was conjectured that the set of decimals whose digits have limiting frequencies p,, pl,..., p9 has fractional dimension-xi pi log,, pi. This was proved by Eggleston (1949). For further brief comments concerning this result, in relation to entropy, see Good (1950b, p. 170) and Shannon (1950, p. 174). An extension of this result to generalized decimals was stated without proof by Good (1951). For generalized decimals see also Everett (1946). The result was further extended to Markov chains of order 1 and proved by Billingsley (1960). His result can be re-expressed in the following less abstract way (in the spirit of generalized decimals). Let the unit interval be broken up into at