Sum of Cantor Sets: Self-similarity and Measure

In this note it is shown that the sum of two homogeneous Cantor sets is often a uniformly contracting self-similar set and it is given a sufficient condition for such a set to be of Lebesgue measure zero (in fact, of Hausdorff dimension less than one and positive Hausdorff measure at this dimension). 1. Definitions and results The study of the arithmetic difference (sum) of two Cantor sets has been of great interest for the homoclinic bifurcations of dynamical systems, since the last decade. For a detailed presentation of this subject we refer the reader to [7]. About 1987, J. Palis made the following Conjecture. If the sum of two affine Cantor sets has positive Lebesgue measure, then it contains an interval. Many articles have been written on this conjecture; see [4], [5], [6] and [8]. Another subject which has been of great interest since the work of B. Mandelbrot ([3]) is the study of fractal dimensions, introduced by Hausdorff in the twenties. The most useful source of examples in this context is given by self-similar sets. Several efforts have been made ([1], [2], [9]) to solve the following Question. Is it true that if a self-similar set has positive Lebesgue measure, then it contains an interval? Our first result (Theorem A below) gives an indication that the conjecture and the question above are related to each other. We are going to state it precisely. Let fi : x ∈ R 7→ cix + ai ∈ R, 0 < ci < 1, i = 1, . . . , r, be affine contractions of the real line. There is a unique compact and nonempty set A ⊂ R such that A = f1(A) ∪ · · · ∪ fr(A) ([2]). This set is called a self-similar set, and it is called a uniformly contracting self-similar set whenever c1 = · · · = cr = c. Let H(2r + 1) be the set of all vectors ~λ = (λ, λ′1, λ, . . . , λ′r, λ) ∈ R, such that λ > 0, λi > 0, i = 1, . . . , r; and λ + λ ′ 1 + λ + · · · + λr + λ = 1. Let C(~λ) be the self-similar set obtained, as above, using the contractions fi(x) = λx + ai, with a0 = 0, ai = λ+λ1 +λ+ · · ·+λi, i = 1, . . . , r. The set C(~λ) is called a homogeneous Cantor set. We have the following Received by the editors February 6, 1998. 1991 Mathematics Subject Classification. Primary 28A78, 58F14.