Long walks in the plane with few collinear points

Joseph Leonide Gerver

Vol. 83 (1979), No. 2, 349–355

DOI: 10.2140/pjm.1979.83.349

Abstract

Let S be a set of vectors in Rⁿ. An S-walk is any (finite or infinite) sequence {z_i} of vectors in Rⁿ such that z_i+1 − z_i ∈ S for all i. We will show that if the elements of S do not all lie on the same line through the origin, then for each integer K ≧ 2, there exists an S-walk W_K = {z_i}_i=1^N(K) such that no K + 1 elements of W_K are collinear and N(K) grows faster than any polynomial function of K.

Mathematical Subject Classification 2000

Primary: 10E99, 10E99

Secondary: 05B99

Milestones

Received: 16 June 1978

Published: 1 August 1979