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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Random intersections of thick Cantor sets


Author: Roger L. Kraft
Journal: Trans. Amer. Math. Soc. 352 (2000), 1315-1328
MSC (1991): Primary 28A80; Secondary 58F99
Published electronically: September 20, 1999
MathSciNet review: 1653359
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Abstract: 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 $C_{1}-C_{2}$.


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Additional Information

Roger L. Kraft
Affiliation: Department of Mathematics, Computer Science and Statistics, Purdue University Calumet, Hammond, Indiana 46323
Email: roger@calumet.purdue.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02464-2
PII: S 0002-9947(99)02464-2
Keywords: Cantor sets, difference sets, thickness
Received by editor(s): October 14, 1997
Published electronically: September 20, 1999
Additional Notes: Research supported in part by a grant from the Purdue Research Foundation
Article copyright: © Copyright 1999 American Mathematical Society