A bound for the representability of large numbers by ternary quadratic forms and nonhomogeneous waring equations

On Waring's problem: some consequences of Golubeva's method

“…This estimate is employed in the argument of the proof of [8,Theorem 2]. We direct the reader to [2, Lemma 1] for a discussion of this conclusion.…”

“…Our goal in this section and the next is the proof of Theorem 1.2, together with the conclusion of Theorem 1.1 which essentially amounts to a corollary of the former theorem. Here we make use of a method originating in work of Linnik [15], and much enhanced by the level-lowering procedure of Golubeva [7,8]. We seek to establish the solubility of the equation (1.4) when n is large by putting x 3 = A + x 0 and x 4 = A − x 0 , with A ∈ N of size nearly n 1/3 and |x 0 | < A.…”

“…Proof. This is in all essentials [8,Theorem 2]. We have modified the conclusion to assert that the integers x, y and z are natural numbers.…”

“…Rather than apply the circle method to establish Theorem 1.1, which as we have noted is limited to situations with β(k) > 2 by the convexity barrier, we instead apply a method of Golubeva involving the theory of ternary quadratic forms [7,8]. In applying this method, we avoid in this paper certain difficulties arising from auxiliary congruences by assuming where necessary the truth of GRH.…”

Fourier coefficients of half-integral weight cusp forms and Waring’s problem

“…Set n " tv 2 w 2 with µ 2 ptq " 1, v N 8 and pw, N q " 1. The square part of n coprime to N , w, can be easily handled by (4)…”