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Polynomial-Time Rademacher Theorem, Porosity and Randomness

Author Alex Galicki

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Alex Galicki

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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)
https://doi.org/10.4230/LIPIcs.ICALP.2017.30

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.
Keywords
  • Rademacher
  • porosity
  • p-randomness
  • differentiability

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