Additive inhomogeneous Diophantine inequalities

New examples of complete sets, with connections to a Diophantine theorem of Furstenberg

“…14. There exists q ≥ 1 such that qα ∈ G. 10 This fact is proven in [19,Theorem 1], but for metric spaces it can be proven more easily as follows: Let G be a compact semigroup of a group, with the group operation written as +. Fix β ∈ G and let (n k ) ∞ 1 be a sequence such that the sequence (n k β) ∞ 1 converges.…”

Sums of cubes with shifts

“…A remark made in the introduction of [5] implies the conclusion of Theorem 1.7 whenever s 13. Theorem 1.7 is obtained in the same way as Theorem 1.1.…”

“…We will require Freeman's bounds on Davenport-Heilbronn minor arcs. In [5,Lemmas 8 and 9], the underlying variables lie in the range (0, P ]. The same results hold, with the same proof, when the underlying variables lie in (bP, cP ] for some fixed real numbers b 0 and c > b.…”

“…An ε-free analogue to Hua's lemma is needed on classical major arcs, and a nontrivial bound is needed on Davenport-Heilbronn minor arcs. The former uses [1,Lemma 4.4], while the latter is provided by [5,Lemmas 8 and 9]. Theorem 1.3 exemplifies the philosophy that if 2t variables suffice to solve an additive problem then t variables suffice almost surely.…”

“…By a simplification of our methods, we may obtain a similar asymptotic formula for sums of five shifted squares. We can also handle sums of nine univariate cubic polynomials, subject to an irrationality condition, improving on the thirteen variable result apparent from Freeman's work [5]. We take the following definition from [5].…”