Introduction to exponential sums and an application to digits
This mini-course is a (gentle) introduction to exponential sums with the aim to give an application to a problem concerning digits of integers. In the first four lectures, I will develop the classical theory, such as Weyl's differencing method, Weyl's criterion and uniform distribution mod 1, the Erdoes-Turan inequality and discrepancy, as well as van der Corput's method. The fifth lecture is a scientific talk: I will show how these classical tools can be used to show that the sum of digits in base 2 and 3 can be "very often very close". This last part is recent joint research with R. de la Bretèche and G. Tenenbaum.