Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Denjoy, Demuth, and Density   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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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: funct-an
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The categories of this paper are: nlin
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The categories of this paper are: comp-gas
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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The categories of this paper are: stat
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The categories of this paper are: nucl-th
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: ao-sci
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The categories of this paper are: supr-con