Transcendence of numbers related to Episturmian words - Kebis, P (2023)
This thesis relates combinatorial properties of sequences to arithmetic properties of the numbers that they represent. A guiding principle is that numbers whose b- expansion has low subword complexity are either rational or transcendental. This heuristic was confirmed by Ferenczi and Maduit, who proved that for b > 1 an inte- ger, numbers whose b-expansion is a Sturmian sequence or an Arnoux-Rauzy sequence are transcendental. (These can be considered as the simplest non-ultimately-periodic sequences in terms of their subword complexity.) Subsequent work of Adamczewski and Bugeaud extended this result by proving that all numbers whose b-expansion has linear subword complexity are rational or transcendental, again for b an integer. The latter authors obtained related results to the case of b an algebraic base under certain combinatorial properties of the sequence, which depend on b.