SELF-SIMILAR SUBSETS OF THE CANTOR SET DE-JUN FENG, HUI RAO, AND YANG WANG Abstract. In this paper, we study the following question raised by Mattila in 1998: what are the self-similar subsets of the middle-third Cantor set C? We give criteria for a complete classification of all such subsets. We show that for any self-similar subset F of C containing more than one point every linear generating IFS of F must consist of similitudes with contraction ratios ±3−n, n 2 N. In particular, a simple criterion is formulated to characterize self-similar subsets of C with equal contraction ratio in modulus. 1. Introduction Let C denote the standard middle-third Cantor set. The main goal of this paper is to answer the following open question raised by Mattila [2] in 1998: what are the self-similar subsets of C? Recall that a non-empty compact set F ⊂ R is said to be self-similar if there k exists a finite family Φ = fφigi=1 of contracting similarity maps on R such that k [ (1.1) F = φi(F): i=1 Such Φ is called a linear iterated function system (IFS) on R. As proved by Hutchin- son [5], for a given IFS Φ, there is a unique non-empty compact set F satisfying (1.1). To specify the relation between Φ and F, we call Φ a linear generating IFS of F, and F the attractor of Φ. Throughout this paper, we use FΦ to denote the attractor of a given linear IFS Φ. A self-similar set F = FΦ is said to be non-trivial if it is not a singleton. The middle-third Cantor set C is one of the most well known examples of self- similar sets. It has a generating IFS fx=3; (x + 2)=3g. Key words and phrases. Cantor set, Self-similar sets, Ternary expansions, Set of uniqueness. 2000 Mathematics Subject Classification: Primary 28A78, Secondary 28A80, 11K16. 1 The first result of this paper is the following theorem, which is our starting point for further investigations. Theorem 1.1. Assume that F ⊆ C is a non-trivial self-similar set, generated by a k linear IFS Φ = fφigi=1 on R. Then for each 1 ≤ i ≤ k, φi has contraction ratio −mi ±3 , where mi 2 N. The proof of the above theorem is based on a short geometric argument and a fundamental result of Salem and Zygmund on the sets of uniqueness in harmonic analysis. k It is easy to see that if a self-similar set F has a generating IFS Φ = fφigi=1 that is derived from the IFS Ψ := fx=3; (x + 2)=3g, i.e. each map in Φ is a finite composition of maps in Ψ, then F ⊆ C. In light of Theorem 1.1, one may guess that each nontrivial self-similar subset of C has a linear generating IFS derived from Ψ. However, this is not true. The following counter example was constructed in [4]. 1 1 1 Example 1.1. Let Φ = f 9 x; 9 (x + 2)g. Choose a sequence (n)n=1 with n 2 f0; 2g P1 −2n+1 so that w = n=1 n3 is an irrational number. Then by looking at the ternary expansion of the elements in FΦ + w := fx + w : x 2 FΦg, it is easy to see that FΦ + w ⊂ C. Observe that FΦ + w is a self-similar subset of C since it is the 0 1 1 attractor of the IFS Φ = f 9 (x + 8w); 9 (x + 2 + 8w)g. However any generating IFS of FΦ0 can not be derived from the original IFS f 0 = x=3; 1 = (x + 2)=3g, since 0 w = min FΦ can not be the fixed point of any map i1i2:::in composed from 0; 1 due to the irrationality of w. The above construction actually shows that C has uncountably many non-trivial self-similar subsets, and indicates the non-triviality of Mattila's question. To further describe self-similar subsets of C, we consider a special class of linear k IFSs. Let F denote the collection of linear IFSs Φ = fφigi=1 on R that satisfy the following conditions (i)-(iii): (i) k ≥ 2 and for each 1 ≤ i ≤ k, φi is of the form −mi (1.2) φi(x) = si3 x + di; si = ±1; mi 2 N; 0 ≤ di ≤ 1: (ii) s1 = 1 and d1 = 0. (iii) di > 0 for at least one 1 ≤ i ≤ k. Moreover di > 0 if si = −1. 2 It is easy to check that for each Φ 2 F, FΦ ⊆ [0; 2] and min FΦ = 0. By the symmetry of C (i.e. C = 1 − C), if F is a non-trivial self-similar subset of C, then 1 − F is also a self-similar subset of C. The following result is a simple consequence of Theorem 1.1 (see Appendix A for a proof). Lemma 1.1. Let F ⊆ C be a non-trivial self-similar set. Then either F or 1 − F can be written as a + E, where a 2 C and E = FΦ for some Φ 2 F. Hence, to characterize the self-similar subsets of C, it is equivalent to characterize the pairs (a; Φ) 2 C × F so that a + FΦ ⊆ C. This is what we will carry out in this paper. We start by introducing some notation and definitions. Notice that any real number x 2 [0; 1] admits at least one ternary expansion 1 X −n x = un3 n=1 with un 2 f0; 2; −2g, and certain rational numbers may admit two different such expansions. For our purpose, we construct a subset Θ ⊂ 0; 2; 2 N, with convention 2 := −2, by n o (1.3) Θ = u 2 0; 2; 2 N : ι(u) 6= −2; u does not end with 221 and 221 ; where 0; if u = 01; (1.4) ι(u) = the first non-zero letter appearing in u; if u 6= 01: Here we say that u ends with c 2 0; 2; 2 N if u = !c for some finite or empty word ! over the alphabet 0; 2; 2 . Then it is easily checked that every x 2 [0; 1] admits P1 −n 1 one and only one ternary expansion n=1 un3 with u = (un)n=1 2 Θ. P1 −n Definition 1.1. For x 2 [0; 1], the unique ternary expansion x = n=1 un3 with 1 (un)n=1 2 Θ, is called the intrinsic ternary expansion of x. P1 −n Clearly any x 2 C admits an intrinsic ternary expansion n=1 bn3 with bn 2 f0; 2g. Now we define the notion of intrinsic translation matrix. −mi k P1 −n Definition 1.2. Let Φ = fφi(x) = si3 x + digi=1 2 F. Let n=1 ui;n3 be the intrinsic ternary expansion of di, i = 1; : : : ; k. Denote by UΦ the following k × 1 3 matrix with entries in 0; 2; 2 : (1.5) UΦ = (ui;n)1≤i≤k;n≥1: We call UΦ the intrinsic translation matrix of Φ. For n 2 N, let UΦ(n) denote the n-th column of UΦ, i.e. T UΦ(n) = (u1;n; : : : ; uk;n) ; where the superscript T denotes the transpose. We say that the vector UΦ(n) is positive if ui;n 2 f0; 2g for all 1 ≤ i ≤ k and uj;n = 2 for at least one j 2 f1; : : : ; kg; correspondingly, we say that UΦ(n) is negative if ui;n 2 f0; 2g for all 1 ≤ i ≤ k and uj;n = 2 for at least one j 2 f1; : : : ; kg. + For m 2 N, let Fm denote the collection of Φ 2 F with uniform contraction ratio −m −m 3 , and Fm the collection of Φ 2 F with contraction ratios ±3 . Now we are ready to present one of our main results. + Theorem 1.2. Let m 2 N and (a; Φ) 2 C × Fm. Define UΦ as in (1.5). Write P1 −n a = n=1 an3 with an 2 f0; 2g. Then a + FΦ ⊆ C if and only if there exists a finite non-empty set M ⊂ N such that the following properties (i)-(iv) hold: (i) Any two numbers in M are incongruent modulo m; (ii) For each n 2 M, UΦ(n) is either positive or negative; T (iii) For each n 2 NnM, UΦ(n) = (0; 0;:::; 0) ; (iv) For any n 2 M, and any integer t ≥ 0, 0; if UΦ(n) is positive; an+mt = 2; if UΦ(n) is negative: + According to the above theorems 1.2, every pair (a; Φ) 2 C × Fm satisfying a + FΦ ⊆ C can be obtained as follows. Choose a finite set M ⊂ N such that any two numbers in M are incongruent modulo m. Let k ≥ 2. Construct an k × 1 matrix U with entries in 0; 2; 2 such that the first row of U consists of the entries 0 only, all the other rows are the sequences in Θ (see (1.3) for the definition of Θ) and moreover, U (instead of UΦ) fulfils the conditions (ii)-(iii) in Theorem 1.2. There are at most 2km different such matrices U once M has been fixed. For i = 1; : : : ; k, let di be the number in [0; 1] with intrinsic ternary expansion given by the i-th row P1 −n of U. Choose a = n=1 an3 2 C such that (an) satisfies (iv) of Theorem 1.2.