Diophantine perspectives to the exponential function and Euler's factorial series

At its simplest, Diophantine approximation is about estimating real numbers with rationals. More generally, one may study a linear form in some given numbers. In case these numbers are such that the linear form is zero only when all the coefficients of the linear form are zero, it is interesting to know how close to zero the linear form can be.

The focus of this thesis is on two functions: the exponential function and Euler's factorial series. Lower bounds for linear forms in the values of these functions are derived using rational function approximations called Padé approximations. The first part of the thesis deals with the exponential function, and the lower bound we obtain gives an improved transcendence measure for Napier's constant.

The construction of Padé approximations leads to large groups of equations whose solutions need to be estimated as well as possible. Such an estimate is given by Siegel's lemma, a fundamental tool in transcendental number theory. Siegel's lemma has later been improved, and the use of this improved version involves finding the greatest common divisor of the maximal minors of the coefficient matrix of the group of equations under consideration. In the second part of the thesis, we investigate the factors of some large determinants related to the use of Siegel's lemma in Padé approximation equations.

In the last part of the dissertation, we consider the factorial series named after Euler which converges in non-Archimedean metrics. We establish some non-vanishing results for a linear form in the values of Euler's series. A lower bound for this linear form is derived as well, improving an earlier corresponding result.