Consider the following questions:
1. How does the volume of the set f(x_1,...,x_d) < epsilon behaves when epsilon goes to 0?
2. How does the number of solutions of the equation f(x_1,...,x_d) = 0 (mod n) behaves when n goes to infinity.
I will present these and other questions which looks as if they are taken from different areas of mathematics. I'll explain the relation between those questions. Then I'll explain how this relation is used in order to show the following theorem answering a question of Larsen and Lubotzky:
For any arithmetic group G there exists a constant C such that the number of irreducible representations of G of dimension n is smaller then C*(n^40).
https://ssl.gstatic.com/ui/v1/icons/mail/images/cleardot.gif