Real quadratic fields with a universal quadratic form of given rank have density zero
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916750061207552 |
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| author | Kala, Vitezslav Yatsyna, Pavlo Żmija, Błażej |
| author_facet | Kala, Vitezslav Yatsyna, Pavlo Żmija, Błażej |
| contents | We prove an explicit upper bound on the number of real quadratic fields that admit a universal quadratic form of a given rank, thus establishing a density zero statement. More generally, we obtain such a result for totally positive definite quadratic lattices that represent all the multiples of a given rational integer. Our main tools are short vectors in quadratic lattices combined with an estimate for the number of periodic continued fractions with bounded coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_12080 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Real quadratic fields with a universal quadratic form of given rank have density zero Kala, Vitezslav Yatsyna, Pavlo Żmija, Błażej Number Theory 11A55, 11E12, 11E20, 11H55, 11R11, 11R80 We prove an explicit upper bound on the number of real quadratic fields that admit a universal quadratic form of a given rank, thus establishing a density zero statement. More generally, we obtain such a result for totally positive definite quadratic lattices that represent all the multiples of a given rational integer. Our main tools are short vectors in quadratic lattices combined with an estimate for the number of periodic continued fractions with bounded coefficients. |
| title | Real quadratic fields with a universal quadratic form of given rank have density zero |
| topic | Number Theory 11A55, 11E12, 11E20, 11H55, 11R11, 11R80 |
| url | https://arxiv.org/abs/2302.12080 |