Sum of Many Dilates

Abstract

We show that for any coprime integers λ 1 , . . . , λ k and any finite A ⊂ Z, one haswhere C only depends on λ 1 , . . . , λ k .

Sums of linear transformations in higher dimensions

“…As previously mentioned, this result was generalised to d-dimensional subsets of Z d by Balog and Shakan in [2]. We refer the reader to [1], [3] and [22] for a more detailed introduction to this problem.…”

“…Bukh gave a lower bound for size of such sets and the main term in Bukh's bound was sharp. The final improvement for Bukh's error term was given by Shakan [22]. As previously mentioned, this result was generalised to d-dimensional subsets of Z d by Balog and Shakan in [2].…”

“…In the case d = 1, a generalization of Theorem 1.1 to sums of several dilates with a better error term was proved by Shakan [22]. Furthermore, when d ≥ 2, previously best known lower bounds for |A + q • A| were by Balog and Shakan [2].…”

“…Similarly, one might be interested in estimates for |A + λ • A| when λ is an algebraic number and A ⊆ Z[λ]. As Shakan remarks in [22,Question 1.2], this is closely related to a conjecture of Bukh that asks for lower bounds for…”

“…In fact, Balog and Shakan give an explicit upper bound for the additive constant C q,s . In [22], Shakan remarks that results like Lemma 2.3 can be extended to A ⊆ R by using a result from [23,Lemma 5.25]. For completeness, we record the same below.…”