Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1 C. Bandt, Self-similar sets 5. integer matrices and fractal tilings of $R^n$, Proc. Amer. Math Soc. 112 (1991), 549--562. MR 92d:58093
  • 2 M. Barnsley, Fractals everywhere, Academic Press, 1988. MR 90e:58080
  • 3 L. Carlitz and L. Moser, On some special factorizations of $(1-X^n)/(1-X)$, Canad. Math. Bull. 9 (1966), 421--426. MR 34:4262
  • 4 W. Gilbert, Geometry of radix expansions, in: The Geometry Vein: the Coxeter Festschrift (1981), 129--139. MR 83j:12001
  • 5 K. Gröchenig and A. Haas, Self--similar lattice tilings, J. Fourier Analysis 1 (1994), 131--170.
  • 6 R. Kenyon, Self-similar tilings, Ph.D thesis, Princeton University (1990).
  • 7 R. Kenyon, Self-replicating tilings, in: Symbolic Dynamics and Applications (P. Walters, ed.) Contemporary Math. vol. 135 (1992), 239--264. MR 94a:52043
  • 8 D. E. Knuth, The art of computer programming: volume 2. Seminumerical algorithms, Addison--Wesley, 1969. (See Chapter 4.1, exercise 20--24.) MR 44:3531
  • 9 J.C. Lagarias and Y. Wang, Self-affine tiles in $\textbf{ % R}^n$, Advances in Math., to appear.
  • 10 J.C. Lagarias and Y. Wang, Integral self-affine tiles in $\textbf{ % R}^n$ I. Standard and nonstandard digit sets, J. London Math. Soc., to appear.
  • 11 D. W. Matula, Basic digit sets for radix representations, J. Assoc. Comput. Mech. 4 (1982), 1131--1143. MR 83k:68017
  • 12 A. M. Odlyzko, Nonnegative digit sets in positional number systems, Proc. London Math. Soc. 37 (1978), 213--229. MR 80m:10004
  • 13 A. Vince, Replicating Tesselations, SIAM J. Discrete Math. 6, no. 3 (1993), 501--521. MR 94e:52023

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11A63, 05B45, 39B42

Retrieve articles in all journals with MSC (1991): 11A63, 05B45, 39B42

Additional Information

Jeffrey C. Lagarias
Affiliation: AT&T Bell Laboratories 600 Mountain Avenue Murray Hill, New Jersey 07974

Yang Wang
Affiliation: School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332

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

American Mathematical Society