On approximation to real numbers by algebraic numbers

On approximation of real numbers by real 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...

Approximation to Real Numbers by Cubic Algebraic Integers I

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 ...

Approximation to real numbers by cubic algebraic integers . II

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 +...

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...