Structure of Tilings of the Line by a Function

A function f 2 L 1 (R) tiles the line with a constant weight w using the discrete tile set A if P a2A f(x \Gamma a) = w almost everywhere. A set A is of bounded density if there is a constant C such that #fa 2 A : n a n + 1g C for all integers n. This paper characterizes compactly supported f 2 L 1 (R) that admit a tiling of R of bounded density. It shows that for such functions all tile sets of bounded density A are finite unions of complete arithmetic progressions. The results apply to some noncompactly supported f 2 L 1 (R). The proofs depend on Cohen's theorem characterizing idempotent measures on locally compact abelian groups. We use a result of Meyer which, using Cohen's theorem, characterizes the collections of point masses on the real line whose Fourier transform is a locally finite measure with total variation that grows at most linearly