Additive Representation in Thin Sequences, Viii: Diophantine Inequalities in Review

The localisation of primes in arithmetic progressions of irrational modulus

“…In the remainder of this section, we briefly indicate why this implies the Theorem. The argument is largely standard, and very similar to the work in Sections 2.1-2.2 of [4]. We are therefore very brief.…”

“…In this section, we establish two estimates by standard applications of the Hardy-Littlewood method. In the proof of the Proposition, they will serve as amplifiers, in the sense of Wooley [10] and Brüdern, Kawada and Wooley [4].…”

Random Diophantine inequalities of additive type

“…We prepare the scene for an application of the Davenport-Heilbronn method, as renovated in [3]. Let 0 < η ≤ 2, and let λ ∈ Λ (s) k .…”

Exceptional Sets for Diophantine Inequalities

“…It would appear that a quantitative version of this result is currently inaccessible, although in the special case where λ 1 /λ 2 is algebraic, such a conclusion has recently been obtained by Brüdern, Kawada and Wooley [10]. In this situation, an explicit bound on the failures of the asymptotic formula is achieved by [10,Theorem 1.5].…”

“…An early preprint of this work existed in mid-2007, with exceptional sets containing power savings in a revision prepared following an Oberwolfach meeting a year later. This work had some influence on the paper on thin sequences [10] joint between Brüdern, Kawada and the second author (see [11] for subsequent developments). With an eye to the delays created by workload obstructions, we have taken the opportunity in this final version of incorporating the very recent developments pertaining to classical Weyl sums stemming from work of the second author on Vinogradov's mean value theorem (see [48], [49] and [50]).…”