Optimal and typical $L^2$ discrepancy of 2-dimensional lattices
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arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866912063240011776 |
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| author | Borda, Bence |
| author_facet | Borda, Bence |
| contents | We undertake a detailed study of the $L^2$ discrepancy of rational and irrational 2-dimensional lattices either with or without symmetrization. We give a full characterization of lattices with optimal $L^2$ discrepancy in terms of the continued fraction partial quotients, and compute the precise asymptotics whenever the continued fraction expansion is explicitly known, such as for quadratic irrationals or Euler's number $e$. In the metric theory, we find the asymptotics of the $L^2$ discrepancy for almost every irrational, and the limit distribution for randomly chosen rational and irrational lattices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_01802 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Optimal and typical $L^2$ discrepancy of 2-dimensional lattices Borda, Bence Number Theory 11K38, 11J83 We undertake a detailed study of the $L^2$ discrepancy of rational and irrational 2-dimensional lattices either with or without symmetrization. We give a full characterization of lattices with optimal $L^2$ discrepancy in terms of the continued fraction partial quotients, and compute the precise asymptotics whenever the continued fraction expansion is explicitly known, such as for quadratic irrationals or Euler's number $e$. In the metric theory, we find the asymptotics of the $L^2$ discrepancy for almost every irrational, and the limit distribution for randomly chosen rational and irrational lattices. |
| title | Optimal and typical $L^2$ discrepancy of 2-dimensional lattices |
| topic | Number Theory 11K38, 11J83 |
| url | https://arxiv.org/abs/2112.01802 |