## Waring’s Problem: Lecture 1

In which we introduce Waring‘s problem and have an overview of how to apply the HardyLittlewood circle method to solve it.

• Waring’s Problem: For any $k\geq 2$, there is $s = s(k)$ such that every positive integer can be written as a sum of $s$ $k^{\textrm{th}}$ powers.  We saw a little of the history of this problem.
• Definition of $g(k)$ and $G(k)$.
• Overview of the asymptotic formula and the way the circle method is used to obtain it, including the idea of major arcs and minor arcs.  We’ll do this in detail over the coming lectures.
• Motto of the course: “If it looks difficult, then approximate!”

Next time, we’ll start the detailed work.  We’ll start by finding an integral that counts the number of solutions to $N = x_1^k + \dotsb + x_s^k$, and then we’ll start the analysis of the minor arcs by thinking about the sum $\sum_{n=0}^{N^{1/k}} e^{2\pi i \theta n^k}$ when $\theta$ is not close to a rational with small denominator (Weyl’s inequality).