Nonnegative Radix Representations for the Orthant $R^n_+$

Abstract:

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