Structure of tilings of the line by a function
type: Item
instance of: scholarly article
title: Structure of tilings of the line by a function
zbMATH Open document ID: 0854.58016
DOI: 10.1215/S0012-7094-96-08227-7
published in: Duke Mathematical Journal
publication date: 1996-07-23T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37E99
Mathematics Subject Classification ID: 37A99
zbMATH DE Number: 908705
zbMATH Keywords: real line
zbMATH Keywords: tilings
zbMATH Keywords: bounded density
author: Mihail N. Kolountzakis
author: Jeffrey C. Lagarias
reviewed by: Mihail N. Kolountzakis
MaRDI profile type: Publication
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full work available at URL: s0012-7094-96-08227-7
OpenAlex ID: W1992241471
description: scientific article; zbMATH DE number 908705