Inventiones mathematicae

, Volume 124, Issue 1, pp 341–365

Tiling the line with translates of one tile

  • Jeffrey C. Lagarias
  • Yang Wang

DOI: 10.1007/s002220050056

Cite this article as:
Lagarias, J. & Wang, Y. Invent math (1996) 124: 341. doi:10.1007/s002220050056

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.

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Jeffrey C. Lagarias
    • 1
  • Yang Wang
    • 2
  1. 1.AT&T Bell Laboratories, Murray Hill, NJ 07974, USA US
  2. 2.Georgia Institute of Technology, Atlanta, GA 30332, USA US