## Abstract

In [GW1] we began an investigation of the following general question. Let *L*
_{1}, . . . , *L*
_{
m
} be a system of linear forms in *d* variables on \({F^n_p}\), and let *A* be a subset of \({F^n_p}\) of positive density. Under what circumstances can one prove that *A* contains roughly the same number of *m*-tuples *L*
_{1}(*x*
_{1}, . . . , *x*
_{
d
}), . . . , *L*
_{
m
}(*x*
_{1}, . . . , *x*
_{
d
}) with \({x_1,\ldots, x_d \in {\mathbb F}^n_p}\) as a typical random set of the same density? Experience with arithmetic progressions suggests that an appropriate assumption is that \({||A - \delta 1||_{U{^k}}}\) should be small, where we have written *A* for the characteristic function of the set *A*, *δ* is the density of *A*, *k* is some parameter that depends on the linear forms *L*
_{1}, . . . , *L*
_{
m
}, and \({|| \cdot ||_U{^k}}\) is the *k*th uniformity norm. The question we investigated was how *k* depends on *L*
_{1}, . . . , *L*
_{
m
}. Our main result was that there were systems of forms where *k* could be taken to be 2 even though there was no simple proof of this fact using the Cauchy–Schwarz inequality. Based on this result and its proof, we conjectured that uniformity of degree *k* − 1 is a sufficient condition if and only if the *k*th powers of the linear forms are linearly independent. In this paper we prove this conjecture, provided only that *p* is sufficiently large. (It is easy to see that some such restriction is needed.) This result represents one of the first applications of the recent inverse theorem for the *U*
^{k} norm over \({F^n_p}\) by Bergelson, Tao and Ziegler [TZ2], [BTZ]. We combine this result with some abstract arguments in order to prove that a bounded function can be expressed as a sum of polynomial phases and a part that is small in the appropriate uniformity norm. The precise form of this decomposition theorem is critical to our proof, and the theorem itself may be of independent interest.

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Both authors gratefully acknowledge the hospitality of the Mathematical Sciences Research Institute, Berkeley, where important parts of this work were carried out.

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Gowers, W.T., Wolf, J. Linear Forms and Higher-Degree Uniformity for Functions On \({\mathbb{F}^{n}_{p}}\)
.
*Geom. Funct. Anal.* **21**, 36–69 (2011). https://doi.org/10.1007/s00039-010-0106-3

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DOI: https://doi.org/10.1007/s00039-010-0106-3