Uniform bounds in Waring’s problem over some diagonal forms

On Waring’s Problem in Sums of Three Cubes for Smaller Powers

“…A naive approach to bounding G 3 (k) would be to replace each sum of three cubes by the specialisation 3x 3 , and this suggests a bound of the shape G 3 (k) ≤ G(3k). With this idea in mind, the bounds G(6) ≤ 24 (due to Vaughan and Wooley [17]), G(9) ≤ 47 and G (12) ≤ 72 (due to Wooley [23]) reveal that our methods improve the trivial approach and confirm that we are actually using the three integral cubes nontrivially in our argument. For the cases k = 2, 3, we combine the pointwise bound obtained for W(α) over the minor arcs with some restriction estimates involving the coefficients a m .…”

“…We observe first that by [16,Theorem 7.1] one has that S y (q, a, b) q 1−1/3k+ε . Moreover, when v ≥ 2 and [12] u = 0 we can deduce from the proof of the same theorem (see in particular the argument following [16,Equation (7.16)]) that S y (p v , a, b) p v−1 . For the case q = p, the work of Weil [19] yields the estimate S y (p, a, b) p 1/2 (see [13, Corollary 2F] for an elementary proof of this bound).…”

“…By applying similar methods we can obtain the same type of approximation for the exponential sum W(α). For a ∈ Z and q ∈ N with (a, q) = 1, and recalling (3-2) and (3-6), we introduce the auxiliary function W(α, q, a) = q −1 y,p S py (q, a)v y,p (β) (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) where y ∈ A(H 3 , P η ) 2 and M/2 ≤ p ≤ M.…”

“…Summing over the range of (y, p) described in (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) delivers the desired result.…”

“…In a recent preprint [12] discussing an analogous problem in which the variables are instead restricted to sums of l th powers, the author focuses on obtaining some uniformity with respect to l in the number of variables needed to attain a representation. The novelty of that paper is a pointwise minor arc estimate for a certain exponential sum which is obtained via an application of the large sieve inequality.…”