Modul: MAT770 Oberseminar: Algebraische Geometrie

## Equidistribution of roots of unity and the Mahler measure

Prof. Dr. Philipp Habegger talk

**Date:** 02.12.19 **Time:** 13.15 - 14.45 **Room:** Y27H25

It is classical that roots of unity of precise order $n$ equidistribute around the unit circle as $n$ tends to infinity. Equidistribution is measured by comparing the average of a continuous test function evaluated at these roots of unity with the integral over the complex unit circle. Baker, Ih, and Rumely extended this to test function with logarithmic singularities of the form $\log|P(\cdot)|$ where $P$ is a univariate polynomial in algebraic coefficients. I will discuss joint work with Vesselin Dimitrov where we allow $P$ to come from a class of a multivariate polynomials. Our method draws from earlier work of Duke. If time permits I will discuss a diophantine application to roots of unity and a connection to a result of Lind, Schmidt, and Verbitskiy on dynamical systems of algebraic origin.