A classical object of study in additive number theory has been the Vinogradov system, that is, the system defined by the equations x1j+ . . . + xsj = y1j+ . . . + ysj (j = 1, . . ., k). In particular, given a finite set A of integers, finding sharp upper bounds for the number of solutions Js,k(A) to this system, when all the variables lie in the set A, has been an important topic of work. Recently, two major approaches have been developed to tackle this problem – the efficient congruencing method of Wooley, and the decoupling techniques of Bourgain-Demeter-Guth. Both these methods give upper bounds for Js,k(A) in terms of s,k, and the cardinality |A| of A, and the diameter X of A. In particular, when X is large in terms of |A|, say when X is much larger than exp(exp(|A|)), these bounds perform worse than the trivial estimates. In this talk, we present new upper bounds for Js,2(A) which depend only on |A| and s. These improve upon, and generalize, a previous result of Bourgain and Demeter.
This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.
