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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Multi-dimensional versions of a theorem of Fine and Wilf and a formula of Sylvester


Authors: R. J. Simpson and R. Tijdeman
Journal: Proc. Amer. Math. Soc. 131 (2003), 1661-1671
MSC (2000): Primary 05D99, 06B25, 11Axx, 11B75, 68R15
Published electronically: January 15, 2003
MathSciNet review: 1953570
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Abstract: Let $ {\vec {v_0},..., \vec {v_k}} $ be vectors in $\mathbf{Z}^k$which generate $\mathbf{Z}^k$. We show that a body $ V \subset \mathbf{Z}^k $ with the vectors $ {\vec {v_0},..., \vec {v_k}} $as edge vectors is an almost minimal set with the property that every function $f: V \rightarrow \mathbf{R}$ with periods $ {\vec {v_0},..., \vec {v_k}} $ is constant. For $k=1$ the result reduces to the theorem of Fine and Wilf, which is a refinement of the famous Periodicity Lemma.

Suppose $ \vec{0} $ is not a non-trivial linear combination of $ {\vec {v_0},..., \vec {v_k}} $ with non-negative coefficients. Then we describe the sector such that every interior integer point of the sector is a linear combination of $ {\vec {v_0},..., \vec {v_k}} $ over $\mathbf{Z}_{\geq 0}$, but infinitely many points on each of its hyperfaces are not. For $k=1$ the result reduces to a formula of Sylvester corresponding to Frobenius' Coin-changing Problem in the case of coins of two denominations.


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Additional Information

R. J. Simpson
Affiliation: Department of Mathematics and Statistics, Curtin University of Technology, P.O. Box U1987, Perth, Western Australia 6001, Australia
Email: simpson@maths.curtin.edu.au

R. Tijdeman
Affiliation: Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands
Email: tijdeman@math.leidenuniv.nl

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06970-3
PII: S 0002-9939(03)06970-3
Keywords: Periodicity, Frobenius, lattice, coin-changing
Received by editor(s): December 31, 2001
Published electronically: January 15, 2003
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2003 American Mathematical Society