The abelian arithmetic regularity lemma
  We give a brief exposition and proof of the arithmetic regularity lemma of Green and Tao in the abelian ($U^2$) case, over $\{1,\dots,N\}$. This may be useful to those who need just the $U^2$ case of the lemma, as the general case is significantly more involved. It may also be useful as an introduction to the general case. No originality is claimed. 
Sean Eberhard
math.NT math.CO