TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 355, Number 4, Pages 1401–1414 S 0002-9947(02)03207-5 Article electronically published on November 20, 2002 ON ONE-DIMENSIONAL SELF-SIMILAR TILINGS AND pq-TILES KA-SING LAU AND HUI RAO Abstract. Let b ≥ 2 be an integer base, D = {0,d 1 , ··· ,d b−1 }⊂ Z a digit set and T = T (b, D) the set of radix expansions. It is well known that if T has nonvoid interior, then T can tile R with some translation set J (T is called a tile and D a tile digit set ). There are two fundamental questions studied in the literature: (i) describe the structure of J ; (ii) for a given b, characterize D so that T is a tile. We show that for a given pair (b, D), there is a unique self-replicating translation set J⊂ Z, and it has period b m for some m ∈ N. This completes some earlier work of Kenyon. Our main result for (ii) is to characterize the tile digit sets for b = pq when p, q are distinct primes. The only other known characterization is for b = p l , due to Lagarias and Wang. The proof for the pq case depends on the techniques of Kenyon and De Bruijn on the cyclotomic polynomials, and also on an extension of the product-form digit set of Odlyzko. 1. Introduction Let T be a compact subset of R with T = T ◦ . If there is a discrete set J⊂ R such that T + J =  t∈J (T + t)= R and {T + t} t∈J is an essentially disjoint family (i.e., (T + t 1 ) ◦ ∩ (T + t 2 ) ◦ = ∅ for any distinct t 1 ,t 2 in J ), then we call T a tile (or a prototile ), J a translation set and (T, J )a (translation) tiling of R. If, further, there is a λ = 0 such that J + λ = J , then we say that (T, J ) is a periodic tiling with period λ. Let b ≥ 2 be an integer, and let D = {d 0 ,d 1 , ··· ,d b−1 } be a subset of R which we call a digit set. The pair (b, D) defines an iterated function system {φ i } b−1 i=0 : φ i (x)= b −1 (x + d i ), 0 ≤ i ≤ b − 1. These maps are contractions, and there is a unique nonempty compact set T = T (b, D) that satisfies the set equation T =  b−1 i=0 φ i (T ) [H]. An equivalent form of the set equation is bT = b−1  i=0 (T + d i )= T + D. (1.1) Received by the editors February 13, 2002 and, in revised form, September 11, 2002. 2000 Mathematics Subject Classification. Primary 52C20, 52C22; Secondary 42B99. The authors are partially supported by an HKRGC grant and also a direct grant from CUHK. The second author is supported by CNSF 19901025. c 2002 American Mathematical Society 1401