Tiling the line with translates of one tile

Summary.

A region $T$ is a closed subset of the real line of positive finite Lebesgue measure which has a boundary of measure zero. Call a region $T$ a tile if${\Bbb R}$ can be tiled by measure-disjoint translates of $T$. For a bounded tile all tilings of${\Bbb R}$ with its translates are periodic, and there are finitely many translation-equivalence classes of such tilings. The main result of the paper is that for any tiling of${\Bbb R}$ by a bounded tile, any two tiles in the tiling differ by a rational multiple of the minimal period of the tiling. From it we deduce a structure theorem characterizing such tiles in terms of complementing sets for finite cyclic groups.