Transcendence of numbers related to Episturmian words

Thesis

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.

The main contribution of this thesis is providing a transcendence result which applies to arbitrary algebraic bases. We introduce a new combinatorial condition on sequences and prove a transcendence result for numbers of the form α := Σ^∞_n=1 u_nβ⁻ⁿ where β is any algebraic number such that |β| > 1 and u = u₁u₂ ··· is a sequence of algebraic numbers satisfying the above-mentioned criterion. In particular we prove that all Episturmian (a generalisation of Arnoux-Rauzy words) words satisfy this criterion.

Authors: Kebis, P

• DOI: 10.5287/ora-xr1pqbnay
• Type of award: MSc
• Level of award: Masters
• Awarding institution: University of Oxford
• Language: English
• Keywords: subspace theorem; Episturmian word; transcendental number; Tribonacci word; Sturmian word; algebraic base
• Subjects: Diophantine approximation