自共形集Khintchine定理的一个类似
An analogue of Khintchine's theorem for self-conformal sets
摘要：
Khintchine定理是度量数论中的一个经典结果，它将某些limsup集的勒贝格测度与自然发生的体积和的收敛/发散联系起来。在本文中，我们问一个类似的结果是否适用于迭代函数系统（IFS）。我们说IFS是近似正则的，如果我们观察到Khintchine型行为，即，如果使用IFS定义的某些limsup集的大小由自然发生的和的收敛/发散确定。证明了一个IFS是逼近正则的，如果它由共形映射组成，并且满足开集条件。我们引入的发散条件包含了IFS中存在的不均匀性。我们通过一个例子证明，这种方法是必不可少的。我们还制定了一个类似的Duffin-Schaeffer猜想，并表明，它持有一套完整的豪斯多夫维数。结合Beresnevich和Velani \cite{BerVel}的质量转移原理，我们证明了一个普遍的结果，这意味着存在例外点内的吸引子我们的IFS。这些点是例外的，因为它们是“非常近似的”。作为这一结果的推论，我们得到了马勒问题的一般解决方案，并证明了有很好的二次无理数逼近的严重。本文件提出的想法是在可能有重叠的综合筹资战略的一般背景下提出的。我们相信，通过从度量数论的角度来看待IFS，人们可以更深入地了解它们重叠的程度。本文的结果应被解释为在这项调查的第一步。
英文摘要：
Khintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurring volume sums. In this paper we ask whether an analogous result holds for iterated function systems (IFSs). We say that an IFS is approximation regular if we observe Khintchine type behaviour, i.e., if the size of certain limsup sets defined using the IFS is determined by the convergence/divergence of naturally occurring sums. We prove that an IFS is approximation regular if it consists of conformal mappings and satisfies the open set condition. The divergence condition we introduce incorporates the inhomogeneity present within the IFS. We demonstrate via an example that such an approach is essential. We also formulate an analogue of the Duffin-Schaeffer conjecture and show that it holds for a set of full Hausdorff dimension. Combining our results with the mass transference principle of Beresnevich and Velani \cite{BerVel}, we prove a general result that implies the existence of exceptional points within the attractor of our IFS. These points are exceptional in the sense that they are "very well approximated". As a corollary of this result, we obtain a general solution to a problem of Mahler, and prove that there are badly approximable numbers that are very well approximated by quadratic irrationals. The ideas put forward in this paper are introduced in the general setting of IFSs that may contain overlaps. We believe that by viewing IFS's from the perspective of metric number theory, one can gain a greater insight into the extent to which they overlap. The results of this paper should be interpreted as a first step in this investigation.
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