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Approximation to Real Numbers by Cubic Algebraic Integers I
Author(s) -
Roy Damien
Publication year - 2004
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s002461150301428x
Subject(s) - mathematics , countable set , rational number , algebraic number , degree (music) , real number , geometry of numbers , diophantine approximation , algebraic extension , discrete mathematics , quadratic equation , combinatorics , mathematical analysis , differential algebraic equation , ordinary differential equation , physics , geometry , diophantine equation , acoustics , minkowski space , differential equation
In 1969, H. Davenport and W. M. Schmidt studied the problem of approximation to a real number ξ 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 ξ and ξ 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 ξ. 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. 2000 Mathematics Subject Classification 11J04 (primary), 11J13, 11J82 (secondary).