Random intersections of thick Cantor sets

Let $C_{1}$, $C_{2}$ be Cantor sets embedded in the real line, and let $\tau _{1}$, $\tau _{2}$ be their respective thicknesses. If $\tau _{1}\tau _{2}>1$, then it is well known that the difference set $C_{1}-C_{2}$ is a disjoint union of closed intervals. B. Williams showed that for some $t\in \operatorname{int}(C_{1}-C_{2})$, it may be that $C_{1}\cap (C_{2}+t)$ is as small as a single point. However, the author previously showed that generically, the other extreme is true; $C_{1}\cap (C_{2}+t)$ contains a Cantor set for all $t$ in a generic subset of $C_{1}-C_{2}$. This paper shows that small intersections of thick Cantor sets are also rare in the sense of Lebesgue measure; if $\tau _{1}\tau _{2}>1$, then $C_{1}\cap (C_{2}+t)$ contains a Cantor set for almost all $t$ in $C_{1}-C_{2}$.