Diophantine approximation and a lower bound for Hausdorff dimension
Diophantine approximation and a lower bound for Hausdorff dimension
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DOI：
10.1112/s0025579300012791
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发表时间：
1990-06
期刊：
影响因子：
0.8
通讯作者：
M. Dodson;B. Rynne;J. Vickers
中科院分区：
文献类型：
--
作者：
M. Dodson;B. Rynne;J. Vickers
集合的一般形式，其中U是一个子集，是一个家庭的子集的U索引的一个集J，是常见的丢番图逼近理论[4，7，18，19]。他们也密切相关的例外集所产生的分析和集的“小因子”在动力系统[1，8，15”。当J是正整数的集合时，集合Λ（λ）当然是集合序列F j的lim-sup，j = 1，2，...[11，p.1]。我们也将把形式（1）的集合称为lim-sup集合，它具有更一般的指数集合J。当这样的lim-sup集的Lebesgue测度为零时，确定它们的Hausdorff维数是很有意义的。通常很难获得Hausdorff维数的良好下限（并且它可能比上限更难确定）。本文对一类相当一般的族给出了形式（1）的lim-sup集的维数的下界，其中包括丢番图逼近理论中的一系列结果。这个下界明确地依赖于集合F α在U中的几何结构和分布。
Sets of the general form where U is a subset of ℛ k and is a family of subsets of U indexed by a set J , are common in the theory of Diophantine approximation [4, 7, 18, 19]. They are also closely connected with exceptional sets arising in analysis and with sets of “small divisors” in dynamical systems [1, 8, 15”. When J is the set of positive integers ℕ, the set Λ(ℱ) is of course the lim-sup of the sequence of sets F j , j = 1, 2,… [11, p. 1]. We will also call sets of the form (1), with the more general index set J, lim-sup sets. When such lim-sup sets have Lebesgue measure zero, it is of interest to determine their Hausdorff dimension. It is usually difficult to obtain a good lower bound for the Hausdorff dimension (and it can be much harder to determine than an upper bound). In this paper we will obtain a lower bound for the dimension of lim-sup sets of the form (1) for a fairly general class of families ℕ which includes a range of results in the theory of Diophantine approximation. This lower bound depends explicitly on the geometric structure and distribution in U of the sets F α in ℕ.