Abstract EN:

This thesis studies links between the combinatorial properties of the base-b expansion or of the continued fraction expansion of a given real number and the fact that this real number is algebraic or transcendental (a conjecture of Borel predicts that every irrational algebraic is an absolutely normal number: its base-b expansions have the properties of a random sequence). We begin with recalling the Ferenczi-Mauduit theorem (combinatorial translation of the Ridout theorem). In 2004, Adamczewski, Bugeaud and Luca appreciably improved this theorem by adapting the Schmidt subspace theorem, proving that irrational algebraics cannot have a complexity with linear growth. In the case of a morphic real number, we establish that this result can be improved and we show that all morphic numbers whose complexity is less than nlog(n) are rational or transcendental. Actually, our result covers even the case of certain morphic numbers with a quadratic complexity. Our condition relates to the growth of the sequence of the sizes of the successive iterations of the first letter under the action of the substitution. Then, this thesis deals with the problematic cases of real numbers x whose base-b expansion is a fixed point of a substitution with polynomial growth. We show that it is still not possible to apply the theorem of Adamczewski-Bugeaud-Luca to rational multiples of the real x. We obtain a similar result when the continued fraction of x is the fixed point of a substitution with polynomial growth, in particular thanks to the algorithm of Raney (which we programmed in Maple language) for computing the continued fraction of a real number x of the image by a rational homography.