On the approximation of algebraic numbers by algebraic integers
نویسندگان
چکیده
منابع مشابه
Algebraic Numbers and Algebraic Integers
c © W W L Chen, 1984, 2013. This chapter originates from material used by the author at Imperial College London between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage an...
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Let n be a positive integer. Let ξ be an algebraic real number of degree greater than n. It follows from a deep result of W. M. Schmidt that, for every positive real number ε, there are infinitely many algebraic numbers α of degree at most n such that |ξ−α| < H(α)−n−1+ε, where H(α) denotes the näıve height of α. We sharpen this result by replacing ε by a function H 7→ ε(H) that tends to zero wh...
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We study effectively the simultaneous approximation of n − 1 different complex numbers by conjugate algebraic integers of degree n over Z( √ −1). This is a refinement of a result of Motzkin [2] (see also [3], p. 50) who has no estimate for the remaining conjugate. If the n−1 different complex numbers lie symmetrically about the real axis, then Z( √ −1) can be replaced by Z. In Section 1 we prov...
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The study of approximation to a real number by algebraic numbers of bounded degree started with a paper of E. Wirsing [10] in 1961. Motivated by this, H. Davenport and W. M. Schmidt considered in [5] the analogous inhomogeneous problem of approximation to a real number by algebraic integers of bounded degree. They proved a result that is optimal for degree 2 and a general result which is valid ...
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It has been conjectured for some time that, for any integer n ≥ 2, any real number ε > 0 and any transcendental real number ξ, there would exist infinitely many algebraic integers α of degree at most n with the property that |ξ−α| ≤ H(α)−n+ε, where H(α) denotes the height of α. Although this is true for n = 2, we show here that, for n = 3, the optimal exponent of approximation is not 3 but (3 +...
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ژورنال
عنوان ژورنال: Journal of the Australian Mathematical Society
سال: 1963
ISSN: 0004-9735
DOI: 10.1017/s1446788700039033