Short proofs of theorems of Lekkerkerker and Ballieu
For any irrational number ξ let A(ξ) denote the set of all accumulation points of {z:z=q(qξ−p), p∈ℤ, q∈ℤ−{0}, p and q relatively prime}. In this paper the following theorem of Lekkerkerker is proved in a short and elementary way: The set A(ξ) is discrete and does not contain zero if and...
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| Format: | Article |
| Language: | English |
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Wiley
1982-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171282000581 |
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| _version_ | 1850224490427973632 |
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| author | Max Riederle |
| author_facet | Max Riederle |
| author_sort | Max Riederle |
| collection | DOAJ |
| description | For any irrational number ξ let A(ξ) denote the set of all accumulation points of {z:z=q(qξ−p), p∈ℤ, q∈ℤ−{0}, p and q relatively prime}. In this paper the following theorem of Lekkerkerker is proved in a short and elementary way: The set A(ξ) is discrete and does not contain zero if and only if ξ is a quadratic irrational. The procedure used for this proof simultaneously takes care of a theorem of Ballieu. |
| format | Article |
| id | doaj-art-4fd003837fe84c7bbb7948948c67da11 |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1982-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-4fd003837fe84c7bbb7948948c67da112025-08-20T02:05:36ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251982-01-015360961210.1155/S0161171282000581Short proofs of theorems of Lekkerkerker and BallieuMax Riederle0Eberhardstr. 14, 79 Ulm/Donau, GermanyFor any irrational number ξ let A(ξ) denote the set of all accumulation points of {z:z=q(qξ−p), p∈ℤ, q∈ℤ−{0}, p and q relatively prime}. In this paper the following theorem of Lekkerkerker is proved in a short and elementary way: The set A(ξ) is discrete and does not contain zero if and only if ξ is a quadratic irrational. The procedure used for this proof simultaneously takes care of a theorem of Ballieu.http://dx.doi.org/10.1155/S0161171282000581 |
| spellingShingle | Max Riederle Short proofs of theorems of Lekkerkerker and Ballieu International Journal of Mathematics and Mathematical Sciences |
| title | Short proofs of theorems of Lekkerkerker and Ballieu |
| title_full | Short proofs of theorems of Lekkerkerker and Ballieu |
| title_fullStr | Short proofs of theorems of Lekkerkerker and Ballieu |
| title_full_unstemmed | Short proofs of theorems of Lekkerkerker and Ballieu |
| title_short | Short proofs of theorems of Lekkerkerker and Ballieu |
| title_sort | short proofs of theorems of lekkerkerker and ballieu |
| url | http://dx.doi.org/10.1155/S0161171282000581 |
| work_keys_str_mv | AT maxriederle shortproofsoftheoremsoflekkerkerkerandballieu |