Approximation of algebraic numbers by algebraic numbers.

On the Approximation to Algebraic Numbers by Algebraic Numbers

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...

On approximation of real numbers by real algebraic numbers

Diophantine approximation by conjugate algebraic numbers

In 1969, Davenport and Schmidt provided upper bounds for the approximation of a real number by algebraic integers. Their novel approach was based on the geometry of numbers and involved the duality for convex bodies. In the present thesis we study the approximation of a real number by conjugate algebraic numbers. We find inspiration in Davenport and Schmidt’s method, but ultimately our approxim...

Approximation of complex algebraic numbers by algebraic numbers of bounded degree

To measure how well a given complex number ξ can be approximated by algebraic numbers of degree at most n one may use the quantities w n (ξ) and w * n (ξ) introduced by Mahler and Koksma, respectively. The values of w n (ξ) and w * n (ξ) have been computed for real algebraic numbers ξ, but up to now not for complex, non-real algebraic numbers ξ. In this paper we compute w n (ξ), w * n (ξ) for a...

Simultaneous Approximation of Logarithms of Algebraic Numbers

Recently, close connections have been established between simultaneous diophantine approximation and algebraic independence. A survey of this topic is given by M. Laurent in these proceedings [7]. These connections are one of the main motivations to investigate systematically the question of algebraic approximation to transcendental numbers. In view of the applications to algebraic independence...