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):= limsupq+∞ 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.