Linear forms of a given Diophantine type and lattice exponents
- Authors: German O.N.1
-
Affiliations:
- Lomonosov Moscow State University
- Issue: Vol 84, No 1 (2020)
- Pages: 5-26
- Section: Articles
- URL: https://bakhtiniada.ru/1607-0046/article/view/133791
- DOI: https://doi.org/10.4213/im8810
- ID: 133791
Cite item
Open Access
Access granted
Subscription Access
Abstract
In this paper we prove an existence theorem concerning linear forms of a given Diophantine type and apply it to study the structure of the spectrum of lattice exponents.
About the authors
Oleg Nikolaevich German
Lomonosov Moscow State University
Email: german.oleg@gmail.com
Doctor of physico-mathematical sciences, no status
References
- Дж. В. Касселс, Введение в геометрию чисел, Мир, М., 1965, 421 с.
- З. И. Боревич, И. Р. Шафаревич, Теория чисел, Наука, М., 1964, 566 с.
- W. M. Schmidt, “Norm form equations”, Ann. of Math. (2), 96:3 (1972), 526–551
- M. M. Skriganov, “Ergodic theory on $operatorname{SL}(n)$, Diophantine approximations and anomalies in the lattice point problem”, Invent. Math., 132:1 (1998), 1–72
- О. Н. Герман, “Диофантовы экспоненты решеток”, Теория чисел и приложения. 1, К 80-летию со дня рождения профессора Анатолия Алексеевича Карацубы, Совр. пробл. матем., 23, МИАН, М., 2016, 35–42
- D. Y. Kleinbock, G. A. Margulis, “Logarithm laws for flows on homogeneous spaces”, Invent. Math., 138:3 (1999), 451–494
- A. Khintchine, “Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen”, Math. Ann., 92:1-2 (1924), 115–125
- N. Technau, M. Widmer, On a counting theorem of Skriganov
- O. N. German, N. G. Moshchevitin, “Linear forms of a given Diophantine type”, J. Theor. Nombres Bordeaux, 22:2 (2010), 383–396
- О. Н. Герман, “Асимптотические направления для наилучших приближений $n$-мерной линейной формы”, Матем. заметки, 75:1 (2004), 55–70
- W. M. Schmidt, L. Summerer, “Parametric geometry of numbers and applications”, Acta Arith., 140:1 (2009), 67–91
- W. M. Schmidt, L. Summerer, “Diophantine approximation and parametric geometry of numbers”, Monatsh. Math., 169:1 (2013), 51–104
- D. Roy, “On Schmidt and Summerer parametric geometry of numbers”, Ann. of Math. (2), 182:2 (2015), 739–786
- A. M. Legendre, Essai sur la theorie des nombres, Duprat, Paris, 1798, xxiv+472 pp.
Supplementary files
