Diophantine approximation by conjugate algebraic numbers

Abstract

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 approximations come from the theory of continued fractions. We get a general optimal result for which we offer two different proofs. We then extend two of Davenport and Schmidt's important results to the context of an imaginary quadratic number field. Our method follows that of Michel Laurent who simplified Davenport and Schmidt's argument in 2003. One of their original results is optimal and so is our extension.