Cantor polynomials for semigroup sectors
Melvyn B. Nathanson

TL;DR
This paper constructs quadratic packing polynomials for certain rational sectors in the plane, proving their existence and uniqueness under specific conditions, and extends the concept to quadratic quasi-polynomial packing functions.
Contribution
It introduces new quadratic packing polynomials for sectors with specific rational slopes and establishes their uniqueness, also extending to quadratic quasi-polynomial packing functions.
Findings
Existence of quadratic packing polynomials for sectors with r dividing s-1
Uniqueness of these polynomials for rational numbers 1/s
Construction of quadratic quasi-polynomial packing functions for all rational sectors
Abstract
A packing function on a set Omega in R^n is a one-to-one correspondence between the set of lattice points in Omega and the set N_0 of nonnegative integers. It is proved that if r and s are relatively prime positive integers such that r divides s-1, then there exist two distinct quadratic packing polynomials on the sector {(x,y) \in \R^2 : 0 \leq y \leq rx/s}. For the rational numbers 1/s, these are the unique quadratic packing polynomials. Moreover, quadratic quasi-polynomial packing functions are constructed for all rational sectors.
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