A little over a decade ago, Ben Green posed the problem of showing that all large integers are the sum of at most a bounded number of bracket quadratic polynomials of the shape n[nθ], for natural numbers n, in which θ is an irrational number such as the square-root of 2. This was resolved in the PhD thesis of Vicky Neale, although no explicit bound was given concerning the number of variables required to achieve success. In this talk we describe a version of Hua’s lemma for this problem that can be applied via the Hardy-Littlewood method to obtain a conclusion with 5 variables. The associated argument differs according to whether θ is a quadratic irrational or not. We also explain how related versions of Hua’s lemma may be interpreted in terms of discrete restriction variants of Vinogradov’s mean value theorem, thus providing a route to generalization.
This video is part of the Number Theory Web Seminar series.
