Dimension of slices through the Sierpinski carpet

Abstract

For Lebesgue typical ( θ , a ) (\theta ,a) , the intersection of the Sierpinski carpet F F with a line y = x tan ⁡ θ + a y=x\tan \theta +a has (if non-empty) dimension s − 1 s-1 , where s = log ⁡ 8 / log ⁡ 3 = dim H ⁡ F s=\log 8/\log 3=\dim _\textrm {H}F . Fix the slope tan ⁡ θ ∈ Q \tan \theta \in \mathbb {Q} . Then we shall show on the one hand that this dimension is strictly less than s − 1 s-1 for Lebesgue almost every a a . On the other hand, for almost every a a according to the angle θ \theta -projection ν θ \nu ^\theta of the natural measure ν \nu on F F , this dimension is at least s − 1 s-1 . For any θ \theta we find a connection between the box dimension of this intersection and the local dimension of ν θ \nu ^\theta at a a .