Linear forms and higher-degree uniformity for functions on $\mathbb{F}_p^n$

Details Linear forms and higher-degree uniformity for functions on $\mathbb{F}_p^n$ Linear forms and higher-degree uniformity for functions on $\mathbb{F}_p^n$

2010-02-10

arxiv.org/abs/1002.2208v1

Mathematics

Number Theory

40 pages

Scientific paper

In [GW09a] we conjectured that uniformity of degree $k-1$ is sufficient to control an average over a family of linear forms if and only if the $k$th powers of these linear forms are linearly independent. In this paper we prove this conjecture in $\mathbb{F}_p^n$, provided only that $p$ is sufficiently large. This result represents one of the first applications of the recent inverse theorem for the $U^k$ norm over $\mathbb{F}_p^n$ by Bergelson, Tao and Ziegler [BTZ09,TZ08]. 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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