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.

Keywords and phrases

Higher order Fourier analysis uniformity norms solutions to systems of linear equations