Summary: Let \(b \geq 2\) be an integer base, \(\mathcal{D} = \{ 0, d_1, \dots , d_{b-1}\} \subset \mathbb{Z}\) a digit set and \(T = T(b, \mathcal{D})\) the set of radix expansions. It is well known that if \(T\) has nonvoid interior, then \(T\) can tile \(\mathbb{R}\) with some translation set \(\mathcal{J}\) (\(T\) is called a tile and \(\mathcal{D}\) a tile digit set). There are two fundamental questions studied in the literature: (i) describe the structure of \(\mathcal{J}\); (ii) for a given \(b\), characterize \(\mathcal{D}\) so that \(T\) is a tile.
We show that for a given pair \((b,\mathcal{D})\), there is a unique self-replicating translation set \(\mathcal{J} \subset \mathbb{Z}\), and it has period \(b^m\) for some \(m \in \mathbb{N}\). 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.