License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2017.30
URN: urn:nbn:de:0030-drops-74033
URL: https://drops.dagstuhl.de/opus/volltexte/2017/7403/
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Galicki, Alex

Polynomial-Time Rademacher Theorem, Porosity and Randomness

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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.

BibTeX - Entry

@InProceedings{galicki:LIPIcs:2017:7403,
  author =	{Alex Galicki},
  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 =	{Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7403},
  URN =		{urn:nbn:de:0030-drops-74033},
  doi =		{10.4230/LIPIcs.ICALP.2017.30},
  annote =	{Keywords: Rademacher, porosity, p-randomness, differentiability}
}

Keywords: Rademacher, porosity, p-randomness, differentiability
Collection: 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)
Issue Date: 2017
Date of publication: 07.07.2017

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