One-dimensional tilings with congruent copies of a 3-point set

摘要

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): = limsup{sub}(q→∞) f(p,q)/q is studied. It is proved that g(2k) = 4/3 or 5/3 and 9(2k+1) = 1 for k ≥ 1.