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

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 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.