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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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