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.
ASJC Scopus subject areas
- Applied Mathematics