Structure of tilings of the line by a function
A function \(f \in L^1(\mathbb{R})\) tiles the line with a constant weight \(w\) using the discrete tile set \(A\) if \(\sum_{a\in A} f(x-a) = w\) almost everywhere. A set \(A\) is of bounded density if there is a constant \(C\) such that \(\#\{a \in A : n \leq a \leq n + 1\} \leq C\) for all integers \(n\). This paper characterizes compactly supported \(f \in L^1 (\mathbb{R})\) that admit a tiling of \(\mathbb{R}\) 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 \(f \in L^1(\mathbb{R})\). 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.
scientific article; zbMATH DE number 908705
scientific article