## An inverse theorem for the Gowers U³(G) norm (2006)

### BibTeX

@MISC{Green06aninverse,

author = {Ben Green and Terence Tao},

title = {An inverse theorem for the Gowers U³(G) norm},

year = {2006}

}

### OpenURL

### Abstract

There has been much recent progress in the study of arithmetic progressions in various sets, such as dense subsets of the integers or of the primes. One key tool in these developments has been the sequence of Gowers uniformity norms U d (G), d = 1, 2, 3,... on a finite additive group G; in particular, to detect arithmetic progressions of length k in G it is important to know under what circumstances the U k−1 (G) norm can be large. The U 1 (G) norm is trivial, and the U 2 (G) norm can be easily described in terms of the Fourier transform. In this paper we systematically study the U 3 (G) norm, defined for any function f: G → C on a finite additive group G by the formula