Approximation to real numbers by cubic algebraic integers I

• Autores: Damien Roy
• Localización: Proceedings of the London Mathematical Society, ISSN 0024-6115, Vol. 88, Nº 1, 2004, págs. 42-62
• Idioma: inglés
• DOI: 10.1112/s002461150301428x
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• Resumen

  • In 1969, H. Davenport and W. M. Schmidt studied the problem of approximation to a real number $\xi$ by algebraic integers of degree at most 3. They did so, using geometry of numbers, by resorting to the dual problem of finding simultaneous approximations to $\xi$ and $\xi^2$ by rational numbers with the same denominator. In this paper, we show that their measure of approximation for the dual problem is optimal and that it is realized for a countable set of real numbers $\xi$. We give several properties of these numbers including measures of approximation by rational numbers, by quadratic real numbers and by algebraic integers of degree at most 3.