Dimension of slices through the sierpinski carpet

Anthony Manning, Károly Simon

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

For Lebesgue typical (θ, a), the intersection of the Sierpinski carpet F with a line y = x tan θ + a has (if non-empty) dimension s - 1, where s = log8/ log 3 = dim H F. Fix the slope tan θ ∈ ℚ. Then we shall show on the one hand that this dimension is strictly less than s - 1 for Lebesgue almost every a. On the other hand, for almost every a according to the angle θ-projection ν θ of the natural measure ν on F, this dimension is at least s-1. For any θ we find a connection between the box dimension of this intersection and the local dimension of ν θ at a.

Original languageEnglish
Pages (from-to)213-250
Number of pages38
JournalTransactions of the American Mathematical Society
Volume365
Issue number1
DOIs
Publication statusPublished - Aug 20 2012

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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