Akshat Mudgal

Diameter free estimates for Vinogradov systems

A classical object of study in additive number theory has been the Vinogradov system, that is, the system defined by the equations
x_1^j+ ... + x_s^j = y_1^j + ...+ y_s^j (1 <= j <= k).
Given a finite set A of integers, finding sharp upper bounds for the number of solutions J_{s,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 J_{s,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 exp(exp(|A|)) << X, these bounds perform worse than the trivial estimates.

In this talk, we present new upper bounds for J_{s,2}(A) which depend only on |A| and s. These improve upon, and generalise, a previous result of Bourgain and Demeter.