Denjoy、Demuth和Density
Denjoy, Demuth, and Density
摘要：
我们考虑两个经典定理的有效版本，勒贝格密度定理和Denjoy-Young-Saks定理。对于第一个，我们证明了一个Martin-Loef随机实$z\in [0,1]$是图灵不完全的当且仅当每个包含$z$的有效闭类$C \subseteq [0,1]$在$z$处有正密度。在更强的假设下，$z$是不是LR-硬，我们证明了$z$有密度1在每个这样的类。这些结果已被应用于解决两个开放的问题之间的相互作用的图灵度的Martin-Loef随机实数和$K$ -平凡集：非杯和覆盖问题。我们说$f\colon[0,1]\to\mathbb{R}$在$z \in [0,1]$满足Denjoy替换，如果导数$f'(z)$存在，或者在$z$的上下导数分别是$+\infty$和$-\infty$。Denjoy-Young-Saks定理指出，每个函数$f\colon[0,1]\to\mathbb{R}$几乎在每个$z\in[0,1]$都满足Denjoy替代。我们回答了Kucera在2004年提出的一个问题表明，一个真正的$z$是可计算的随机当且仅当每个可计算的功能$f$满足Denjoy替代$z$。对于马尔可夫可计算函数，它只定义在可计算实数上，我们可以使用伪导数来制定Denjoy替代方案。如果每个马尔可夫可计算函数在$z$满足Denjoy替换，则称一个实$z$ DA-随机。我们大大加强了Demuth（评论）的结果。卡罗琳大学，24（3）：391- 406，1983），证明了每个图灵不完全Martin-Loef随机实数是DA-随机的。证明涉及到非孔隙度的概念，密度的变体，这是本文的两个主题之间的桥梁。最后，我们证明了DA-随机性与Martin-Loef随机性是不可比拟的。
英文摘要：
We consider effective versions of two classical theorems, the Lebesgue density theorem and the Denjoy-Young-Saks theorem. For the first, we show that a Martin-Loef random real $z\in [0,1]$ is Turing incomplete if and only if every effectively closed class $C \subseteq [0,1]$ containing $z$ has positive density at $z$. Under the stronger assumption that $z$ is not LR-hard, we show that $z$ has density-one in every such class. These results have since been applied to solve two open problems on the interaction between the Turing degrees of Martin-Loef random reals and $K$-trivial sets: the non-cupping and covering problems. We say that $f\colon[0,1]\to\mathbb{R}$ satisfies the Denjoy alternative at $z \in [0,1]$ if either the derivative $f'(z)$ exists, or the upper and lower derivatives at $z$ are $+\infty$ and $-\infty$, respectively. The Denjoy-Young-Saks theorem states that every function $f\colon[0,1]\to\mathbb{R}$ satisfies the Denjoy alternative at almost every $z\in[0,1]$. We answer a question posed by Kucera in 2004 by showing that a real $z$ is computably random if and only if every computable function $f$ satisfies the Denjoy alternative at $z$. For Markov computable functions, which are only defined on computable reals, we can formulate the Denjoy alternative using pseudo-derivatives. Call a real $z$ DA-random if every Markov computable function satisfies the Denjoy alternative at $z$. We considerably strengthen a result of Demuth (Comment. Math. Univ. Carolin., 24(3):391--406, 1983) by showing that every Turing incomplete Martin-Loef random real is DA-random. The proof involves the notion of non-porosity, a variant of density, which is the bridge between the two themes of this paper. We finish by showing that DA-randomness is incomparable with Martin-Loef randomness.
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