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

Abstract

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.

40 pages

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• Structure of finite nilspaces and inverse theorems for the Gowers norms in bounded exponent groups
• Polynomial bound for the partition rank vs the analytic rank of tensors
• On the Schmidt and analytic ranks for trilinear forms
• A Gap in the Subrank of Tensors
• An inverse theorem for Freiman multi-homomorphisms
• Structure and regularity for subsets of groups with finite VC-dimension
• A quantitative inverse theorem for the $U^4$ norm over finite fields
• Partition and Analytic Rank are Equivalent over Large Fields
• The inverse conjecture for the Gowers norm over finite fields in low characteristic
• Uniformity of multiplicative functions and partition regularity of some quadratic equations
• Induced arithmetic removal: complexity 1 patterns over finite fields
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• True complexity of polynomial progressions in finite fields
• Sparse graph counting and Kelley-Meka bounds for binary systems
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• True complexity and iterated Cauchy--Schwarz
• Further bounds in the polynomial Szemerédi theorem over finite fields
• Relative Rank and Regularization
• Higher-order Fourier analysis of ${\mathbb{F}}_p^n$ and the complexity of systems of linear forms
• Biased multilinear maps of abelian groups
• Additive combinatorics with a view towards computer science and cryptography: An exposition
• Induced arithmetic removal for partition-regular patterns of complexity 1
• Correlation Testing for Affine Invariant Properties on ${\mathbb{F}}_p^n$ in the High Error Regime
• Ranges of polynomials control degree ranks of Green and Tao over finite prime fields
• Local aspects of the Sidorenko property for linear equations