Sums of Three Cubes

Breaking classical convexity in Waring's problem: Sums of cubes and quasi-diagonal behaviour

“…The problems of providing lower bounds for 9 ! '(X), and upper bounds for E(X), have long histories (see, for example, [2,3,4,5,6,7,9,II,12,13]). Theorem 1.2 permits us to make some further, very small, progress.…”

Sums of three cubes, II

“…Thirty-five years later, Vaughan [19], [20] enhanced these methods, first proving that N(X) ≫ X 8/9−ε , and later that N(X) ≫ X 19/21−ε . His seminal introduction [21] of methods utilising smooth numbers led to the lower bound N(X) ≫ X 11/12−ε (see also Ringrose [18] for an intermediate result). The author's derivation of effective estimates for fractional moments of smooth Weyl sums [24] first delivered a lower bound of the shape N(X) ≫ X 1−ξ/3−ε , where ξ = 0.24956813 .…”