Cantor polynomials for semigroup sectors

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.