Linear forms from the Gowers uniformity norm

Abstract

This is a companion note to our paper 'A relative Szemerédi theorem', elaborating on a concluding remark. In that paper, we showed how to prove a relative Szemerédi theorem for (r + 1)-term arithmetic progressions assuming a linear forms condition. Here we show how to replace this condition with an assumption about the Gowers uniformity norm U^r.

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