On one-dimensional self-similar tilings and $pq$-tiles

Let $b \geq 2$ be an integer base, $\mathcal{D} = \{ 0, d_1, \cdots , d_{b-1}\} \subset \mathbb{Z}$ 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.