Continued fractions with sequences of partial quotients
Continued fractions with sequences of partial quotients
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DOI：
10.1090/s0002-9939-1973-0311581-4
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发表时间：
1973-02
期刊：
影响因子：
--
通讯作者：
K. Hirst
中科院分区：
文献类型：
--
作者：
K. Hirst
证明了关于部分商属于给定序列的连分式集的Hausdorff分数维的三个结果。导言。在[1]中，I.J.Good研究了部分商满足各种条件的连分式集合的分数维。在这些结果中包含了讨论An变大的情况的定理，在[2]中，这些结果被扩展到包括An迅速趋于无穷大的情况。在所有这些结果中，对An的唯一限制是An_f(N)和An--oo类型。在这篇文章中，我将证明关于An被进一步限制为属于某个自然数序列的情况的类似结果。连分式和Hausdorff测度理论的符号和相关部分在[2]中给出了，读者可以参考那篇论文来了解这些细节。此外，在整篇文章中，我们将使用(Bn)来表示严格递增的自然数序列(而不是(+(N))-以便于打印)。将证明以下定理。定理1.设级数E(On)-‘收敛。定理2.假设级数E(Bn)-‘发散，则对任一A，定理1中的集合E具有1ac维.
Three results are proved concerning the Hausdorff fractional dimension of sets of continued fractions whose partial quotients belong to given sequences. Introduction. In [1], I. J. Good investigated the fractional dimension of sets of continued fractions whose partial quotients an obey various conditions. Included amongst these results are theorems discussing cases where an becomes large, and in [2] these results were extended to cover some cases where an tends to infinity rapidly. In all these results the only restrictions on an are of the type an_f(n) and an--oo. In this paper I shall prove analogous results concerning the cases where an is further restricted to belong to some sequence of natural numbers. The notation to be used and the relevant parts of the theories of continued fractions and Hausdorff measures are given in [2] and the reader is referred to that paper for these details. In addition throughout the paper we shall use (bn) to denote a strictly increasing sequence of natural numbers (rather than (+(n))-for ease of printing). The following theorems will be proved. THEOREM 1. Suppose the series E (on)-' converges. Then provided A has the property that >,(on)" -A the set E={jai.>A and ai E (bn)} has fractional dimension _ ?cx. THEOREM 2. Suppose the series E (bn)-' diverges. Then for any A, the set E in Thleorem 1 has dimension 1ac.