One-dimensional Tilings Using Tiles with Two Gap Lengths

Abstract.

Let n,p,k,q,l be positive integers with n=k+l+1. Let x₁,x₂, . . . ,x _n be a sequence of positive integers with x₁<x₂<···<x _n . A set {x₁,x₂, . . . ,x _n } is called a set of type (p,k;q,l) if the set of differences {x₂−x₁,x₃−x₂, . . . ,x _n −x _{n −1}} equals {p, . . . ,p,q, . . . ,q} as a multiset, where p and q appear k and l times, respectively. Among other results, it is shown that for any p,k,q, there exists a finite interval I in the set of integers such that I is partitioned into sets of type (p,k;q,1).