Waring’s Problem: Lecture 12

In which we take a brief look at a variant of Waring’s Problem.

• There were no details in this lecture, but we had a quick look at how to go about writing numbers as sums of bracket quadratics $n \lfloor n\sqrt{2} \rfloor$ rather than $k^{\mathrm{th}}$ powers.

Davenport’s book (Analytic methods for Diophantine equations and Diophantine inequalities) and Vaughan’s book (The Hardy-Littlewood method) both have numerous applications of the circle method to other problems (some involving $k^{\mathrm{th}}$ powers, some involving inequalities rather than equations, and some involving primes, for example).  In particular, Vaughan’s book discusses Vinogradov’s three primes theorem, a spectacular success of the method, and both books discuss further techniques for improving the bounds in Waring’s problem.  There have recently been some significant improvements in these bounds, proved by Trevor Wooley; you can find the relevant papers on his website.  Terry Tao has just written an interesting blog post about the limitations of the circle method.