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DOI: 10.4230/LIPIcs.ICALP.2017.30

Polynomial-Time Rademacher Theorem, Porosity and Randomness

Authors: Alex Galicki

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Abstract

The main result of this paper is a polynomial time version of Rademacher's theorem. We show that if z is p-random, then every polynomial time computable Lipschitz function f:R^n->R is differentiable at z. This is a generalization of the main result of [Nies, STACS2014]. To prove our main result, we introduce and study a new notion, p-porosity, and prove several results of independent interest. In particular, we characterize p-porosity in terms of polynomial time computable martingales and we show that p-randomness in R^n is invariant under polynomial time computable linear isometries.

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Alex Galicki. Polynomial-Time Rademacher Theorem, Porosity and Randomness. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 30:1-30:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{galicki:LIPIcs.ICALP.2017.30,
  author =	{Galicki, Alex},
  title =	{{Polynomial-Time Rademacher Theorem, Porosity and Randomness}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{30:1--30:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.30},
  URN =		{urn:nbn:de:0030-drops-74033},
  doi =		{10.4230/LIPIcs.ICALP.2017.30},
  annote =	{Keywords: Rademacher, porosity, p-randomness, differentiability}
}

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Polynomial-Time Rademacher Theorem, Porosity and Randomness

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