Arithmetic properties of exceptional lattice paths
For a fixed real number ρ > 0, let L be an affine line of slope ρ ⁻¹ in R ² . We show that the closest approximation of L by a path P in Z ² is unique, except in one case, up to integral translation. We study this exceptional case. For irrational ρ, the projection of P to L yields two q...
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| Date: | 2006 |
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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут прикладної математики і механіки НАН України
2006
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| Series: | Algebra and Discrete Mathematics |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/157386 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Arithmetic properties of exceptional lattice paths / W. Rump // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 3. — С. 101–118. — Бібліогр.: 16 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | For a fixed real number ρ > 0, let L be an affine
line of slope ρ
⁻¹
in R
²
. We show that the closest approximation of
L by a path P in Z
²
is unique, except in one case, up to integral
translation. We study this exceptional case. For irrational ρ, the
projection of P to L yields two quasicrystallographic tilings in the
sense of Lunnon and Pleasants [5]. If ρ satisfies an equation x
² =
mx + 1 with m ∈ Z, both quasicrystals are mapped to each other
by a substitution rule. For rational ρ, we characterize the periodic
parts of P by geometric and arithmetic properties, and exhibit
a relationship to the hereditary algebras Hρ(K) over a field K
introduced in a recent proof of a conjecture of Ro˘ıter. |
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