The Thue-Morse word over the alphabet {a, b} is the fixed point starting with a of the morphism sending a to ab and b to ba, that is, it is the limit of the sequence of finite words a, ab, abba, abbabaab, abbabaabbaababba, … We survey the Diophantine properties of real numbers whose expansion in some integer base or whose continued fraction expansion is given by a Thue-Morse word (here, a and b are distinct positive integers). We also discuss the Diophantine properties of p-adic numbers whose Hensel expansion is a Thue-Morse word and of power series over a finite field F whose continued fraction expansion is a Thue-Morse word (here, a and b are distinct nonconstant polynomials with coefficients in F).
I will talk about a new equidistribution result for the roots of polynomials with bounded Bombieri norm. This can be formulated and proven in a much broader setting for the zero sets of small sections of hermitian line bundles on arithmetic varieties in any dimension. I will discuss the main ingredients of the proof coming from geometry of numbers and from complex analysis. This is work in progress.
If one wants to treat integration of differential forms over semi-algebraic sets analogous to the case of smooth manifolds, it is desirable to have triangulations of semi-algebraic sets that are globally of class $C^1$. We will present a proof of the existence of such triangulations using the 'panel beating' method introduced by Ohmoto-Shiota (2017) and discuss possible generalizations.
