On sets of natural numbers whose difference set contains no squares
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We show that if a sequence s/ of natural numbers has no pair of elements whose difference is a positive square, then the density of J/ n{l,...,«} is O(l/log«) c »), cn->-oo. This improves previous results which showed that the density converges to zero, but at a slower rate. We use a technique based on the method of Hardy and Littlewood together with a combinatorial result that is of independent interest. The approach may be useful for other problems in additive number theory.
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