Nonnegative radix representations for the orthant R₊ⁿ

Let A be a nonnegative real matrix which is expanding, i.e. with all eigenvalues |λ| > 1, and suppose that |det(A)| is an integer. Let D consist of exactly |det(A)| nonnegative vectors in Rⁿ. We classify all pairs (A, D) such that every x in the orthant R₊ⁿ has at least one radix expansion in base A using digits in 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 D can be diagonally scaled to lie in Zⁿ. The proofs generalize a method of Odlyzko, previously used to classify the one-dimensional case.