Translating the Cantor set by a random real
Translating the Cantor set by a random real
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将康托尔集平移为随机实数
DOI：
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发表时间：
2014
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通讯作者：
Jason Teutsch
中科院分区：
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作者：
R. Dougherty;Jack H. Lutz;R. Mauldin;Jason Teutsch
我们确定了康托集合随机平移中点的构造维数。在某种意义上，康托集合“取消了随机性”，当它的一些成员被添加到马丁-洛夫随机实数时，识别出一个比随机本身具有更低构造维数的点。特别地，我们找到了在给定构维的康托集合中点的集合平移的豪斯多夫维数。1. 分形与随机实数我们探索算法随机性、经典分形几何和加性数论之间的基本相互作用。在本文中，我们考虑了一个给定集合与另一个给定集合的平移交点的维数。我们不仅要关注经典的豪斯多夫测度和维数，还要关注这些概念的有效类比。更具体地说，设C表示标准的中三分康托集[7,18]，对于每个数α，设（1.1）E=α = {x: cdimH{x} =α}由所有具有构维数α的实数组成。我们通过证明以下定理来回答道格·哈丁提出的问题：定理1.1。如果1−log 2/ log 3≤α≤1，且r是Martin-Lof随机实数，则（1.2）(C + r)∩E=α的Hausdorff维数为α−(1−log 2/ log 3)。此外，该集合的Hausdorff测度在其维度上是正的。从这个结果我们得到了（1.2）的有效和经典Hausdorff维数之间的简单关系；差就是1减去康托集的维数。我们得出结论，在康托集合中的许多点加性地抵消了随机性。我们讨论了本文涉及的一些概念。直观地说，如果一个实数不从一个有效的null类继承任何特殊属性，那么它就是“随机的”。如果一个数字“通过”了所有MartinLof测试，我们就说它是Martin-Lof random[3,13]。马丁-洛夫检验是一个统一可计算的数列。2000数学学科分类。主68 q30;二级11K55, 28A78。
We determine the constructive dimension of points in random translates of the Cantor set. The Cantor set “cancels randomness” in the sense that some of its members, when added to Martin-Lof random reals, identify a point with lower constructive dimension than the random itself. In particular, we find the Hausdorff dimension of the set of points in a Cantor set translate with a given constructive dimension. 1. Fractals and random reals We explore an essential interaction between algorithmic randomness, classical fractal geometry, and additive number theory. In this paper, we consider the dimension of the intersection of a given set with a translate of another given set. We shall concern ourselves not only with classical Hausdorff measures and dimension but also the effective analogs of these concepts. More specifically, let C denote the standard middle third Cantor set [7, 18], and for each number α let (1.1) E=α = {x : cdimH{x} = α} consist of all real numbers with constructive dimension α. We answer a question posed to us by Doug Hardin by proving the following theorem: Theorem 1.1. If 1− log 2/ log 3 ≤ α ≤ 1 and r is a Martin-Lof random real, then the Hausdorff dimension of (1.2) (C + r) ∩ E=α is α−(1− log 2/ log 3). Moreover the Hausdorff measure of this set in its dimension is positive. From this result we obtain a simple relation between the effective and classical Hausdorff dimensions of (1.2); the difference is exactly 1 minus the dimension of the Cantor set. We conclude that many points in the Cantor set additively cancel randomness. We discuss some of the notions involved in this paper. Intuitively, a real is “random” if it does not inherit any special properties by belonging to an effective null class. We say a number is Martin-Lof random [3, 13] if it “passes” all MartinLof tests. A Martin-Lof test is a uniformly computably enumerable (c.e.) sequence 2000 Mathematics Subject Classification. Primary 68Q30; Secondary 11K55, 28A78.