On Sets of Natural Numbers Whose Difference Set Contains No Squares

Abstract

We show that if a sequence A of natural numbers has no pair of elements whose difference is a positive square, then the density of A ∩{l,…, n} is O ( 1 log ⁡ n c n ) , cn → ∞. 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.