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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Jeffrey C. Lagarias and Yang Wang
Journal: Trans. Amer. Math. Soc. 348 (1996), 99-117
MSC (1991): Primary 11A63; Secondary 05B45, 39B42
MathSciNet review: 1333392
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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 $\R^n$. We classify all pairs $(A, {\mathcal D})$ such that every $x$ in the orthant $\R^n_+$ 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 $\Z^n$. The proofs generalize a method of Odlyzko, previously used to classify the one--dimensional case.


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Additional Information

Jeffrey C. Lagarias
Affiliation: AT&T Bell Laboratories 600 Mountain Avenue Murray Hill, New Jersey 07974
Email: jcl@research.att.com

Yang Wang
Affiliation: School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332
Email: wang@math.gatech.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01538-3
PII: S 0002-9947(96)01538-3
Received by editor(s): July 1, 1994
Additional Notes: Research supported in part by the National Science Foundation, grant DMS–9307601
Article copyright: © Copyright 1996 American Mathematical Society