Abstract
It is proved that given e>0, there is δ(e)>0 such that ifS is a measurable set of [0,N], |S|>eN, then there is a triplex, x+h, x+h2 inS withh satisfyingh>δ(e)N1/2. The argument is related to [B] and uses the behavior of certain non-linear convolution-type operators. The method can be adapted in a variety of situations. For instance, it can be used to prove the analogue of the previous statement with the square replaced by another power,h3,h4 etc.
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