Dimension of slices through the sierpinski carpet

Anthony Manning, K. Simon

Research output: Contribution to journalArticle

15 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 language English 213-250 38 Transactions of the American Mathematical Society 365 1 https://doi.org/10.1090/S0002-9947-2012-05586-3 Published - Aug 20 2012

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Sierpinski Carpet
Slice
Henri Léon Lebésgue
tan(x+y)
Intersection
Local Dimension
Box Dimension
Slope
Strictly
Projection
Angle
Line

ASJC Scopus subject areas

• Mathematics(all)
• Applied Mathematics

Cite this

Dimension of slices through the sierpinski carpet. / Manning, Anthony; Simon, K.

In: Transactions of the American Mathematical Society, Vol. 365, No. 1, 20.08.2012, p. 213-250.

Research output: Contribution to journalArticle

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