Packing Polynomials on Multidimensional Integer Sectors

Denoting the real numbers and the nonnegative integers, respectively, by R and N, let S be a subset of Nn for n = 1, 2, . . ., and f be a mapping from Rn into R. We call f a packing function on S if the restriction f |S is a bijection onto N. For all positive integers r1, . . . , rn−1, we consider the integer sector I(r1, . . . , rn−1) = {(x1, . . . , xn) ∈ Nn | xi+1 6 rixi for i = 1, . . . , n − 1}. Recently, Melvyn B. Nathanson (2014) proved that for n = 2 there exist two quadratic packing polynomials on the sector I(r). Here, for n > 2 we construct 2n−1 packing polynomials on multidimensional integer sectors. In particular, for each packing polynomial on Nn we construct a packing polynomial on the sector I(1, . . . , 1).