Tiling the line with translates of one tile. (English) Zbl 0847.05037

In this paper the tilings are of the real line \(\mathbb{R}\) by translates of a single compact tile \(T\) of positive Lebesgue measure: \[ \mathbb{R} = \bigcup_{t \in {\mathcal T}} (T + t). \] The intersection of the interiors of any two distinct tiles must be empty. Such tilings can be subtle in the sense that the tile \(T\) can have even infinitely many connected components; a large class of such tiles arises from self-similar constructions. The discrete set \(\mathcal T\) is called the tiling set. A tiling is said to be periodic if \(\mathcal T = {\mathcal T} + \lambda\) for some \(\lambda \in \mathbb{R} \setminus \{0\}\). The minimum such \(\lambda\) is called the period. The first result is that every such tiling must be periodic and the period must be an integral multiple of the Lebesgue measure \(\mu(T)\). The second result, using Fourier theoretic methods, concerns the cosets of such a periodic tiling. It states that, if the tiling set is given by \[ {\mathcal T} = \bigcup^J_{j = 1} (r_j + \lambda \mathbb{Z}), \] then all differences \(r_j - r_k\) are rational multiples of the period \(\lambda\). The analogous statements, of both results, for higher dimensions are false. The second result is applied to obtain a structure theorem, in terms of complementing sets for finite cyclic groups, for possible tiles \(T\) that give such tilings.


05B45 Combinatorial aspects of tessellation and tiling problems
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