Dimension of slices through the Sierpinski carpet

Abstract:

For Lebesgue typical $(\theta ,a)$, the intersection of the Sierpinski carpet $F$ with a line $y=x\tan \theta +a$ has (if non-empty) dimension $s-1$, where $s=\log 8/\log 3=\dim _\textrm {H}F$. Fix the slope $\tan \theta \in \mathbb {Q}$. 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 $\theta$-projection $\nu ^\theta$ of the natural measure $\nu$ on $F$, this dimension is at least $s-1$. For any $\theta$ we find a connection between the box dimension of this intersection and the local dimension of $\nu ^\theta$ at $a$.