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Recently, I had tentatively announced a forthcoming result with Ben Green establishing the “Gowers inverse conjecture” (or more accurately, the “inverse conjecture for the Gowers uniformity norm”) for vector spaces {\Bbb F}_p^n over a finite field {\Bbb F}_p, in the special case when p=2 and when the function f: {\Bbb F}_p^n \to {\Bbb C} for which the inverse conjecture is to be applied is assumed to be a polynomial phase of bounded degree (thus f= e^{2\pi i P/|{\Bbb F}|}, where P: {\Bbb F}_p^n \to {\Bbb F}_p is a polynomial of some degree d=O(1)). See my FOCS article for some further discussion of this conjecture, which has applications to both polynomiality testing and to various structural decompositions involving the Gowers norm.

This conjecture can be informally stated as follows. By iterating the obvious fact that the derivative of a polynomial of degree at most d is a polynomial of degree at most d-1, we see that a function P: {\Bbb F}_p^n \to {\Bbb F}_p is a polynomial of degree at most d if and only if

\sum_{\omega_1,\ldots,\omega_{d+1} \in \{0,1\}} (-1)^{\omega_1+\ldots+\omega_{d+1}} P(x +\omega_1 h_1 + \ldots + \omega_{d+1} h_{d+1}) = 0

for all x,h_1,\ldots,h_{d+1} \in {\Bbb F}_p^n. From this one can deduce that a function f: {\Bbb F}_p^n \to {\Bbb C} bounded in magnitude by 1 is a polynomial phase of degree at most d if and only if the Gowers norm

\|f\|_{U^{d+1}({\Bbb F}_p^n)} := \bigl( {\Bbb E}_{x,h_1,\ldots,h_{d+1} \in {\Bbb F}_p^n} \prod_{\omega_1,\ldots,\omega_{d+1} \in \{0,1\}}

{\mathcal C}^{\omega_1+\ldots+\omega_{d+1}} f(x + \omega_1 h_1 + \ldots + \omega_{d+1} h_{d+1}) \bigr)^{1/2^{d+1}}

is equal to its maximal value of 1. The inverse conjecture for the Gowers norm, in its usual formulation, says that, more generally, if a function f: {\Bbb F}_p^n \to {\Bbb C} bounded in magnitude by 1 has large Gowers norm (e.g. \|f\|_{U^{d+1}} \geq \varepsilon) then f has some non-trivial correlation with some polynomial phase g (e.g. \langle f, g \rangle > c(\varepsilon) for some c(\varepsilon) > 0). Informally, this conjecture asserts that if a function has biased (d+1)^{th} derivatives, then one should be able to “integrate” this bias and conclude that the function is biased relative to a polynomial of degree d. The conjecture has already been proven for d \leq 2. There are analogues of this conjecture for cyclic groups which are of relevance to Szemerédi’s theorem and to counting linear patterns in primes, but I will not discuss those here.

At the time of the announcement, our paper had not quite been fully written up. This turned out to be a little unfortunate, because soon afterwards we discovered that our arguments at one point had to go through a version of Newton’s interpolation formula, which involves a factor of d! in the denominator and so is only valid when the characteristic p of the field exceeds the degree. So our arguments in fact are only valid in the range p > d, and in particular are rather trivial in the important case p=2; my previous announcement should thus be amended accordingly.

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