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!”

Further reading

I mentioned some resources at the start of the lecture.

  • Davenport Analytic Methods for Diophantine Equations and Diophantine Inequalities (CUP).  A very readable introduction to the Hardy-Littlewood circle method.
  • Vaughan The Hardy-Littlewood Method (CUP).  Lots of useful information.
  • Iwaniec and Kowalski Analytic Number Theory (AMS).  A large book with a small (but useful) section on the circle method.
  • Ben Green’s lecture notes from a Part III course on Additive Number Theory (Chapter 3 is particularly relevant).

If you have suggestions of your own, please do leave them in a comment below.

Preparation for Lecture 2

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).


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