William L. Steiger

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We show that if a sequence s/ of natural numbers has no pair of elements whose difference is a positive square, then the density of J / n{l , . . . ,«} is O(l/log«)»), cn->-oo. This improves previous results which showed that the density converges to zero, but at a slower rate. We use a technique based on the method of Hardy and Littlewood together with a(More)
Given disjoint sets PI, P2 . . . . . Pd in R a with n points in total, a hamsandwich cut is a hyperplane that simultaneously bisects the Pi. We present algorithms for finding ham-sandwich cuts in every dimension d > 1. When d = 2, the algorithm is optimal, having complexity O(n). For dimension d > 2, the bound on the running time is proportional to the(More)
We discuss several generalizations of the notion of median to points in $R^d$. They arise in Computational Geometry and in Statistics. The notions are compared with respect to some of their mathematical properties. We also consider computational aspects. The issue of computational complexity raises several intriguing questions.
We consider three problems about the illumination of planar regions with oodlights of prescribed angles. Problem 1 is the decision problem: given a wedge W of angle , n points p 1 ; ; p n in the plane and n angles 1 ; ; n summing up to , decide whether W can be illuminated by oodlights of angles 1 ; ; n placed in some order at the points p 1 ; ; p n and(More)
Abstract Let S be a set with n = w+ b points in general position in the plane, w of them white, and b of them black, where w and b are even numbers. We show that there exists a matching of points of the same color with straight line segments and no crossings which matches at least 83:33% of the points. We also derive an O(n logn) algorithm which achieves(More)