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 ex...
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| Veröffentlicht in: | Algebra and Discrete Mathematics |
|---|---|
| Datum: | 2006 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут прикладної математики і механіки НАН України
2006
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/157386 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Arithmetic properties of exceptional lattice paths / W. Rump // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 3. — С. 101–118. — Бібліогр.: 16 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862536065655504896 |
|---|---|
| author | Rump, W. |
| author_facet | Rump, W. |
| citation_txt | Arithmetic properties of exceptional lattice paths / W. Rump // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 3. — С. 101–118. — Бібліогр.: 16 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | 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.
|
| first_indexed | 2025-11-24T11:09:58Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-157386 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-11-24T11:09:58Z |
| publishDate | 2006 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Rump, W. 2019-06-20T03:11:02Z 2019-06-20T03:11:02Z 2006 Arithmetic properties of exceptional lattice paths / W. Rump // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 3. — С. 101–118. — Бібліогр.: 16 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 05B30, 11B50; 52C35, 11A0 https://nasplib.isofts.kiev.ua/handle/123456789/157386 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. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Arithmetic properties of exceptional lattice paths Article published earlier |
| spellingShingle | Arithmetic properties of exceptional lattice paths Rump, W. |
| title | Arithmetic properties of exceptional lattice paths |
| title_full | Arithmetic properties of exceptional lattice paths |
| title_fullStr | Arithmetic properties of exceptional lattice paths |
| title_full_unstemmed | Arithmetic properties of exceptional lattice paths |
| title_short | Arithmetic properties of exceptional lattice paths |
| title_sort | arithmetic properties of exceptional lattice paths |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/157386 |
| work_keys_str_mv | AT rumpw arithmeticpropertiesofexceptionallatticepaths |