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On sum-intersective sets. (English) Zbl 0709.11013
A set H of integers is called sum-intersective if for every set A of positive upper density the equation \(a+a'=h\) has a solution with \(a,a'\in A\), \(h\in H\). “Natural” sets do not have this property, since such a set must intersect every arithmetic progression of even numbers. Probabilistic arguments show the existence of a sum-intersective set satisfying \(H(x)=\sum_{h\in H,h\leq x}1\ll (\log x)^{2+\epsilon}\) [the reviewer, Astérisque 147/148, 173-182 (1986; Zbl 0625.10046)], but constructions had more than \(x^ c\) elements. In this paper the following thinner construction is given. Let \[ H=\{mp^ k :\;m\leq p,k\geq 0.2 (\log p)^{1/3}(\log \log p)^{-2/3}\}, \] where p runs over prime numbers. This set, whose counting function is \[ H(x)<\exp c(\log x)^{3/4}(\log \log x)^{1/2}, \] is proved to be sum-intersective. The proof is based on the Hardy-Littlewood-Vinogradov method.
Reviewer: I.Z.Ruzsa
MSC:
11B13 Additive bases, including sumsets
11P32 Goldbach-type theorems; other additive questions involving primes
11P55 Applications of the Hardy-Littlewood method
11B34 Representation functions
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