Polynomial indexing of integer lattice-points I. General concepts and quadratic polynomials

Abstract

Denoting the nonnegative integers by N and the signed integers by Z, we let S be a subset of Zm for m = 1, 2,... and f be a mapping from S into N. We call f a storing function on S if it is injective into N, and a packing function on S if it is bijective onto N. Motivation for these concepts includes extendible storage schemes for multidimensional arrays, pairing functions from recursive function theory, and, historically earliest, diagonal enumeration of Cartesian products. Indeed, Cantor's 1878 denumerability proof for the product N2 exhibits the equivalent packing functions fCantor(x, y) = {either x or y} + (x + y)(x + y + 1) 2 on the domain N2, and a 1923 Fueter-Pólya result, in our terminology, shows fCantor the only quadratic packing function on N2. This paper extends the preceding result. For any real-valued function f on S we define a density S ÷ f = limn→∞ ( 1 n)#{S {pitchfork} f-1([-n, +n])}, and for any packing function f on S we observe the fact S ÷ f = 1. Using properties of this density, and invoking Davenport's lemma from geometric number theory, we find all polynomial storing functions with unit density on N, and exclude any polynomials with these properties on Z, then find all quadratic storing functions with unit density on N2, and exclude any quadratics with these properties on Z × N, Z2. The admissible quadratics on N2 are all nonnegative translates of fCantor. An immediate sequel to this paper excludes some higher-degree polynomials on subsets of Z2. © 1978.