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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