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The inverse conjecture for the Gowers norm over finite fields

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Let F\mathbb{F} be the finite field under consideration, let VV be a finite-dimensional vector space over F\mathbb{F}, and let Us+1(V)U^{s+1}(V) denote the Gowers norm. A function f:V→Cf:V\to\mathbb{C} is 1-bounded if ∣f(x)∣⩽1|f(x)|\leqslant 1 for every x∈Vx\in V. For a polynomial P:V→TP:V\to\mathbb{T}, write

e(−P(x))=exp⁡(−2πiP(x)).e(-P(x))=\exp(-2\pi iP(x)).

Inverse conjecture GI⁡(s)\operatorname{GI}(s). Let δ>0\delta>0 and s⩾0s\geqslant 0. Then there exists an ε=εδ,s,F>0\varepsilon=\varepsilon_{\delta,s,\mathbb{F}}>0 such that for every finite-dimensional vector space VV and any 1-bounded function f:V→Cf:V\to\mathbb{C} with

∣f∣Us+1(V)⩾δ,\\|f\\|_{U^{s+1}(V)}\geqslant\delta,

there exists P∈Poly⁡⩽s(V→T)P\in\operatorname{Poly}_{\leqslant s}(V\to\mathbb{T}) such that

∣Ex∈Vf(x)e(−P(x))∣⩾ε.\left|\mathbb{E}_{x\in V}f(x)e(-P(x))\right|\geqslant\varepsilon.

This is the inverse statement to the fact that correlation with a polynomial phase of degree at most s+1s+1 implies a large Us+1U^{s+1} norm. The conjecture asks for quantitative correlation with a phase of degree at most ss in low characteristic; the supplied text does not state whether it has been proved or disproved.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Inverse conjecture for the Gowers norm over finite fields

    Let F\mathbb{F} be a finite field, let VV be a finite-dimensional vector space over F\mathbb{F}, and let D:={z∈C:∣z∣⩽1}\mathcal{D}:= \{z \in \mathbb{C}: |z| \leqslant 1\} be the compact unit disk. For an integer d⩾1d \geqslant 1, write ∥f∥Ud(V)\|f\|_{U^{d}(V)} for the Gowers norm and ∥f∥ud(V)\|f\|_{u^{d}(V)} for the weak Gowers norm of a function f:V→Df:V\to\mathcal{D}. Inverse conjecture for the Gowers norm. For every δ>0\delta>0 there exists ε>0\varepsilon>0 such that

    ∥f∥ud(V)⩾ε\|f\|_{u^{d}(V)} \geqslant \varepsilon

    for every finite vector space VV and every function f:V→Df:V\to\mathcal{D} such that

    ∥f∥Ud(V)⩾δ.\|f\|_{U^{d}(V)} \geqslant \delta.

    This formulation fails in the low-characteristic regime char⁡(F)+1<d\operatorname{char}(\mathbb{F})+1<d, although the cited counterexamples do not rule out the formulation above in that case because of the distinction between phase polynomials and exponentials of ordinary polynomials. The relationship between the Gowers and weak Gowers norms is therefore particularly subtle in low characteristic.

    source: Terence Tao and Tamar Ziegler, “The inverse conjecture for the Gowers norm over finite fields via the correspondence principle”, arXiv:0810.5527 (2009).

References

Primary source

Terence Tao and Tamar Ziegler, “The inverse conjecture for the Gowers norm over finite fields in low characteristic”, arXiv:1101.1469 (2011).

additive-combinatoricsfinite-fieldsgowers-norms
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