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- W1976337900 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." @default.
- W1976337900 created "2016-06-24" @default.
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- W1976337900 date "2011-01-08" @default.
- W1976337900 modified "2023-10-11" @default.
- W1976337900 title "Linear Forms and Higher-Degree Uniformity for Functions On $${mathbb{F}^{n}_{p}}$$" @default.
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- W1976337900 doi "https://doi.org/10.1007/s00039-010-0106-3" @default.
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