# Linear Forms and Higher-Degree Uniformity for Functions On \({\mathbb{F}^{n}_{p}}\)

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## 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.

## Keywords and phrases

Higher order Fourier analysis uniformity norms solutions to systems of linear equations## 2010 Mathematics Subject Classification

11B30## Preview

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