Random Intersections of Thick Cantor Sets

Let C1, C2 be Cantor sets embedded in the real line, and let τ1, τ2 be their respective thicknesses. If τ1τ2 > 1, then it is well known that the difference set C1 − C2 is a disjoint union of closed intervals. B. Williams showed that for some t ∈ int(C1−C2), it may be that C1∩ (C2 + t) is as small as a single point. However, the author previously showed that generically, the other extreme is true; C1 ∩ (C2 + t) contains a Cantor set for all t in a generic subset of C1 − C2. This paper shows that small intersections of thick Cantor sets are also rare in the sense of Lebesgue measure; if τ1τ2 > 1, then C1 ∩ (C2 + t) contains a Cantor set for almost all t in C1 − C2. If C1, C2 are Cantor sets embedded in the real line, then their difference set is C1 − C2 ≡ { x− y | x ∈ C1 and y ∈ C2 }. The difference set has another, more dynamical, definition as C1 − C2 = { t | C1 ∩ (C2 + t) 6= ∅ }, where C2 + t = { x + t | x ∈ C2 } is the translation of C2 by the amount t. There are two reasons to say that the second definition is dynamical. First, it gives a dynamic way of visualizing the difference set; if we think of C1 as being fixed in the real line and think of C2 as sliding across C1 with unit speed, then C1 − C2 can be thought of as giving those times when the moving copy of C2 intersects C1. Second, it has become a tool for studying dynamical systems. One Cantor set sliding over another one comes up in various studies of homoclinic phenomena, such as infinitely many sinks, [N1], antimonotonicity, [KKY], and Ω-explosions, [PT1]; for an elementary explanation of this, see [GH, pp. 331–342] or [R, pp. 110–115]. This has led to a number of problems and results of the following form: Given conditions on the sizes of C1 and C2, what can be said of the sizes of either C1−C2, or C1∩(C2 + t) for t ∈ C1−C2. A wide variety of notions of size have been used, such as cardinality, topology, measure, Hausdorff dimension, limit capacity, and thickness; see for example [HKY], [KP], [MO], [PT2], [PS], [S], and [W]. In this paper we will be concerned with the thickness of C1 and C2, and our conclusion will be about the topology of C1 ∩ (C2 + t) for almost every t ∈ C1 − C2. It is not hard to show that the difference set of two Cantor sets C1, C2 is always a compact, perfect set. So the simplest structure that we can expect C1 − C2 to have is the disjoint union of closed intervals. There is a condition we can put on C1 and C2 that will guarantee this; if τ1, τ2 are the thicknesses of C1, C2, and Received by the editors October 14, 1997. 1991 Mathematics Subject Classification. Primary 28A80; Secondary 58F99.