Partitio Numerorum: sums of squares and higher powers

Abstract

We survey the potential for progress in additive number theory arising from recent advances concerning major arc bounds associated with mean value estimates for smooth Weyl sums. We focus attention on the problem of representing large positive integers as sums of a square and a number of $k$-th powers. We show that such representations exist when the number of $k$-th powers is at least $\lfloor c_0k\rfloor +2$, where $c_0=2.136294\ldots $. By developing an abstract framework capable of handling sequences with appropriate distribution properties, analogous conclusions are obtained, for example, when the square is restricted to have prime argument.