Arithmetic Harmonic Analysis 2021 - Lecture 1 скачать в хорошем качестве

Arithmetic Harmonic Analysis 2021 - Lecture 1

This is the first lecture in the course. In it, I begin our discussion of Waring's problem focussed on sums of squares. After stating the theorem, I give a simple `divisor argument' that yields an upper bound that is almost the correct size. From there, I use the Vinogradov's approach to write the represent the delta function as an oscillatory integral. Using this, the number of respresentations is expressed as the inverse Fourier transform of an exponential sum. From here, I give an overview of the proof. The rest of the lecture and the next two (Lectures 2 and 3) are devoted to analyzing this exponential sum, peeling away error terms until we reach the main term.