A reduction technique in Waring's problem, I

Minor Arc Moments of Weyl Sums

“…We note that the logarithmic power saving is of the same strength of Boklan [2] when k = 3. We also note that for k ≥ 6, Boklan [3] derived an asymptotic formula for Waring's problem in 7 8 2 k variables by saving a power of a logarithm in the minor arc integral.…”

“…In order to estimate n≤x r(n) 2 , we shall study the analytic properties of the Dirichlet series 2 n s , which converges for s > 1. Recall that the Dedekind zeta function of K is defined to be Let E/F be a finite Galois extension of number fields, with Galois group G = Gal(E/F).…”

Additive representation in thin sequences, III: asymptotic formulae

“…, whence the asymptotic formula (1.1) holds for almost all n. Indeed, by employing refinements due to Boklan [1], one may show that any δ < 3 is permissible. The only available conclusion of which we are aware for thin sequences is due to Brüdern and Watt [6], who demonstrate the validity of the formula (1.1) for R 4 (n) for almost all integers n in certain short intervals.…”

“…The mean value I 1 is swiftly estimated by reference to Boklan [1], who established the upper bound (2.9)…”

Rational points on linear slices of diagonal hypersurfaces

“…Fix a number θ with 2 1−k k < θ ≤ 4 5 and then take Q = P θ . Let m denote the set of real numbers α ∈ [0, 1] for which the inequality |qα − a| ≤ QP −k with q ∈ N, a ∈ Z is only possible when q > Q. due to Vaughan (see [26], Theorem B, for k = 3 and [27] for k ≥ 4, with a slightly different power of log P ; for refinements see Boklan [2] and Harvey [14]). Vaughan's approach involves a considerable refinement of the conventional proof of Hua's Lemma (see [29], Lemma 2.5).…”