TILING THE LINE WITH TRANSLATES OF ONE TILE

A region T is a closed subset of the real line of positive finite Lebe sgue measure which has a boundary of measure zero. Call a region T a t ile if R can be tiled by measure-disjoint translates of T. For a bound ed tile all tilings of R with its translates are periodic, and there a re finitely many translation equivalence classes of such tilings. The main result of the paper is that for any tiling of R by a bounded tile , any two tiles in the tiling differ by a rational multiple of the min imal period of the tiling. From it we deduce a structure theorem chara cterizing such tiles in terms of complementing sets for finite cyclic groups.