Counting algebraic numbers in short intervals with rational points
In 2012 it was proved that real algebraic numbers follow a nonuniform but regular distribution, where the respective definitions go back to H. Weyl (1916) and A. Baker and W. Schmidt (1970). The largest deviations from the uniform distribution occur in neighborhoods of rational numbers with small d...
| Main Authors: | , , |
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| Format: | Article |
| Language: | Belarusian |
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Belarusian State University
2019-04-01
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| Series: | Журнал Белорусского государственного университета: Математика, информатика |
| Subjects: | |
| Online Access: | https://journals.bsu.by/index.php/mathematics/article/view/923 |
| _version_ | 1818065414132858880 |
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| author | Vasily I. Bernik Friedrich Götze Nikolai I. Kalosha |
| author_facet | Vasily I. Bernik Friedrich Götze Nikolai I. Kalosha |
| author_sort | Vasily I. Bernik |
| collection | DOAJ |
| description | In 2012 it was proved that real algebraic numbers follow a nonuniform but regular distribution, where the respective definitions go back to H. Weyl (1916) and A. Baker and W. Schmidt (1970). The largest deviations from the uniform distribution occur in neighborhoods of rational numbers with small denominators. In this article the authors are first to specify a gene ral condition that guarantees the presence of a large quantity of real algebraic numbers in a small interval. Under this condition, the distribution of real algebraic numbers attains even stronger regularity properties, indicating that there is a chance of proving Wirsing’s conjecture on approximation of real numbers by algebraic numbers and algebraic integers. |
| first_indexed | 2024-12-10T14:51:30Z |
| format | Article |
| id | doaj.art-94c2c11311fa4fd08b2f0977b13fc769 |
| institution | Directory Open Access Journal |
| issn | 2520-6508 2617-3956 |
| language | Belarusian |
| last_indexed | 2024-12-10T14:51:30Z |
| publishDate | 2019-04-01 |
| publisher | Belarusian State University |
| record_format | Article |
| series | Журнал Белорусского государственного университета: Математика, информатика |
| spelling | doaj.art-94c2c11311fa4fd08b2f0977b13fc7692022-12-22T01:44:26ZbelBelarusian State UniversityЖурнал Белорусского государственного университета: Математика, информатика2520-65082617-39562019-04-01141110.33581/2520-6508-2019-1-4-11923Counting algebraic numbers in short intervals with rational pointsVasily I. Bernik0Friedrich Götze1Nikolai I. Kalosha2Institute of Mathematics, National Academy of Sciences of Belarus, 11 Surhanava Street, Minsk 220072, BelarusBielefeld University, 25 Universitätsstraße, Bielefeld D33615, GermanyInstitute of Mathematics, National Academy of Sciences of Belarus, 11 Surhanava Street, Minsk 220072, BelarusIn 2012 it was proved that real algebraic numbers follow a nonuniform but regular distribution, where the respective definitions go back to H. Weyl (1916) and A. Baker and W. Schmidt (1970). The largest deviations from the uniform distribution occur in neighborhoods of rational numbers with small denominators. In this article the authors are first to specify a gene ral condition that guarantees the presence of a large quantity of real algebraic numbers in a small interval. Under this condition, the distribution of real algebraic numbers attains even stronger regularity properties, indicating that there is a chance of proving Wirsing’s conjecture on approximation of real numbers by algebraic numbers and algebraic integers.https://journals.bsu.by/index.php/mathematics/article/view/923algebraic numberdiophantine approximationuniform distributiondirichlet’s theoremkhinchine’s theorem |
| spellingShingle | Vasily I. Bernik Friedrich Götze Nikolai I. Kalosha Counting algebraic numbers in short intervals with rational points Журнал Белорусского государственного университета: Математика, информатика algebraic number diophantine approximation uniform distribution dirichlet’s theorem khinchine’s theorem |
| title | Counting algebraic numbers in short intervals with rational points |
| title_full | Counting algebraic numbers in short intervals with rational points |
| title_fullStr | Counting algebraic numbers in short intervals with rational points |
| title_full_unstemmed | Counting algebraic numbers in short intervals with rational points |
| title_short | Counting algebraic numbers in short intervals with rational points |
| title_sort | counting algebraic numbers in short intervals with rational points |
| topic | algebraic number diophantine approximation uniform distribution dirichlet’s theorem khinchine’s theorem |
| url | https://journals.bsu.by/index.php/mathematics/article/view/923 |
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