Random intersections of thick Cantor sets

Abstract

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