On lattice point weak b-visibility
Allie Aird
Alberto Alonso
Samuel I Cooper
Patrick M Crossley
Pamela E Harris
George Kuliner
Robert J Marino
Eric S Piato
Barbara J Schweitzer
SUNY Geneseo
Reuben Kaufman
Abstract
For a fixed b ∈ Z+ , a point (r, s) ∈ Z × Z is b-visible from the origin if there exists a power function f (x) = axb with a ∈ Q such that f(0)=0 and f(r)=s, and no other point in the integer lattice belongs to the graph of f. In this article, we extend the definition of b-visibility given by Goins, Harris, Kubik, and Mbirika to the study of weak visibility. For a fixed b ∈ Z+ , we say that a point Q = (h, k) in the array ∆m,n = {1, 2, . . . , m} × {1, 2, . . . , n} is weakly b-visible from a point P = (r, s) ∈ Z+ × Z+ such that P ∈ ∆m,n if no other point in ∆m,n lies on the curve f(x)= ((s−k)/(r−h)b) (x − h)b + k between Q and P . In this paper we give necessary and sufficient conditions for determining if a point in ∆m,n is weakly b-visible by an external point. We also show that for any point P = (r, s) with r > m and s > n, there exists a b ≥ 1 such that every point in ∆m,n is weakly b-visible from P . Our last result considers a fixed b > 1 and specifies the coordinates of a point P that weakly b-views every point in ∆m,n, and as a corollary we provide a way to determine the coordinates of the closest point to the array satisfying such a condition. We conclude by providing a few directions for future research.
