One-Dimensional Tilings with Congruent Copies of a 3-Point Set

Abstract

For given positive integers p and q, let f(p,q) be the smallest integer n such that {0,1,...,3n – 1} can be partitioned into congruent copies of a 3-point set {0,p, p + q}. It is shown that f(p,q) is approximately at most 5q/3 for any fixed p and large q. Moreover, g(p) := lim sup _q→ ∞ f(p,q)/q is studied. It is proved that g(2k) = 4/3 or 5/3 and g(2k + 1) = 1 for k ≥ 1.