On certain sequences of lattice points

Joseph Leonide Gerver and Lawrence Thom Ramsey

Vol. 83 (1979), No. 2, 357–363

DOI: 10.2140/pjm.1979.83.357

Abstract

Let S be a finite subset of Rⁿ. A sequence {z_i} is an S-walk if and only if z_i+1 −z_i is an element of S for all i. In an effective manner it is shown that long S-walks in Z² must have an increasing number of collinear points. In Z³, however, an infinite S-walk may have a bounded number of collinear points.

Mathematical Subject Classification 2000

Primary: 10E99, 10E99

Secondary: 05B99

Milestones

Received: 28 February 1978

Published: 1 August 1979