Periodicity of tilings by sets of finite measure on the real line

Let Ω ⊂ ℝ be measurable, with 0 < |Ω| < ∞. We prove that every ordinary translational tiling of the line by Ω has a periodic translation set. For a fixed tile, all such translation sets have finitely many translation classes and share a common period T with T/|Ω| ∈ ℕ. The set Ω need not be bounded. The main analytic input is a smooth compactly supported window inside the nonzero set of the Fourier transform of 1_Ω on which the exponential sums associated with every packing by Ω satisfy uniform Riesz inequalities. This window gives uniqueness for tiling complements which approach one another at one end. A continuous circle-valued flux on the compact space of tiling complements then reduces periodicity to a finite expansive system. The theorem answers the level-one indicator question posed by Kolountzakis and Lev.