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On sets of natural numbers whose difference set contains no squares. (English) Zbl 0651.10031
Let $$A$$ denote any sequence of natural numbers. Let $$A_n$$ denote $$A\cap \{1,2,\ldots,n\}$$. Let $$\sigma =\sigma_n$$ denote the number of natural numbers in $$A_n$$. Let $$d(A_n)$$ denote the density $$\sigma/n$$ of $$A_n$$. For any two sets $$B$$ and $$C$$ of natural numbers let $$B-C$$ denote all natural numbers of the form $$b-c$$ where $$b$$ is in $$B$$ and $$c$$ is in $$C$$. Then the authors prove the following theorem: There exist positive constants $$c_0$$ and $$c_1$$, such that for any sequence $$A$$ such that for $$n>c_1$$, $$A_n-A_n$$ is free from squares we have $d(A_n)<c_0(\log n)^{-(1/12)(\log \log \log \log n)}.$ This theorem is a very nice and deep improvement on earlier results. The argument uses the method of Hardy and Littlewood together with a combinatorial result which is of independent interest.

##### MSC:
 11P55 Applications of the Hardy-Littlewood method 11B13 Additive bases, including sumsets 11B83 Special sequences and polynomials
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