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

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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 kth 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 kth 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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Correspondence to J. Wolf.

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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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