Nonnegative Radix Representations for the Orthant 𝑅ⁿ₊

Abstract

Let A A be a nonnegative real matrix which is expanding, i.e. with all eigenvalues | λ | > 1 |\lambda | > 1 , and suppose that | det ( A ) | |\det (A)| is an integer. Let D {\mathcal D} consist of exactly | det ( A ) | |\det (A)| nonnegative vectors in R n \mathbb {R}^n . We classify all pairs ( A , D ) (A, {\mathcal D}) such that every x x in the orthant R + n \mathbb {R}^n_+ has at least one radix expansion in base A A using digits in D {\mathcal D} . The matrix A A must be a diagonal matrix times a permutation matrix. In addition A A must be similar to an integer matrix, but need not be an integer matrix. In all cases the digit set D \mathcal D can be diagonally scaled to lie in Z n \mathbb {Z}^n . The proofs generalize a method of Odlyzko, previously used to classify the one–dimensional case.