Transcendence of Sturmian Numbers over an Algebraic Base

Abstract: Ferenczi and Mauduit showed in 1997 that a number represented over an integer base by a Sturmian sequence of digits is transcendental. In this talk we generalise this result to hold for all algebraic number base b of absolute value strictly greater than one. More generally, for a given base b and given irrational number θ, we prove rational linear independence of the set comprising 1 together with all numbers of the above form whose associated digit sequences have slope θ.

We give an application of our main result to the theory of dynamical systems. We show that for a Cantor set C arising as the set of limit points of a contracted rotation f on the unit interval, where f is assumed to have an algebraic slope, all elements of C except its endpoints 0 and 1 are transcendental.

This is joint work with Florian Luca and Joel Ouaknine.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

---

One World Numeration seminar

Series comments: Description: Online seminar on numeration systems and related topics

For questions or subscribing to the mailing list, contact the organisers at numeration@irif.fr

---