Optimal exponent for some problems in Diophantine Approximation

An important aspect of Diophantine Approximation deals with the problem of approximating real or complex numbers by rational numbers or, more generally, by algebraic numbers of bounded degree. This study provides criteria to decide whether a given real or complex number is algebraic or transcendental. In this thesis we present several such results. Following Davenport & Schmidt we look at the approximation of a real number by rational numbers, by quadratic irrational numbers and by algebraic integers of degree at most 3. We also look at the related problem of simultaneous approximation of a real number and its square by rational numbers with the same denominator. We conclude with a new Gel'fond type criterion in degree 2 and show that it involves an optimal exponent of approximation.