Lipschitz constant log{n} almost surely suffices for mapping n grid points onto a cube

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Abstract

Kaluža, Kopecká and the author have shown that the best Lipschitz constant for mappings taking a given nd-element set in the integer lattice ℤd, with n∈ℕ, surjectively to the regular n times n grid {1,…,n}d may be arbitrarily large. However, there remain no known, non-trivial asymptotic bounds, either from above or below, on how this best Lipschitz constant grows with n. We approach this problem from a probabilistic point of view. More precisely, we consider the random configuration of nd points inside a given finite lattice and establish almost sure, asymptotic upper bounds of order log n on the best Lipschitz constant of mappings taking this set surjectively to the regular n times n grid {1,…,n}d.
Original languageEnglish
PublisherarXiv
DOIs
Publication statusPublished - 28 Oct 2020

Bibliographical note

18 pages

Keywords

  • math.FA
  • 51F99, 51M05, 52C99, 26B10

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