Nonnegative Radix Representations for the Orthant $R^n_+$
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- by Jeffrey C. Lagarias and Yang Wang PDF
- Trans. Amer. Math. Soc. 348 (1996), 99-117 Request permission
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.References
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Additional Information
- Jeffrey C. Lagarias
- Affiliation: AT&T Bell Laboratories 600 Mountain Avenue Murray Hill, New Jersey 07974
- MR Author ID: 109250
- Email: jcl@research.att.com
- Yang Wang
- Affiliation: School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332
- Email: wang@math.gatech.edu
- Received by editor(s): July 1, 1994
- Additional Notes: Research supported in part by the National Science Foundation, grant DMS–9307601
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 99-117
- MSC (1991): Primary 11A63; Secondary 05B45, 39B42
- DOI: https://doi.org/10.1090/S0002-9947-96-01538-3
- MathSciNet review: 1333392