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In Notes 3, we saw that the number of additive patterns in a given set was (in principle, at least) controlled by the Gowers uniformity norms of functions associated to that set.

Such norms can be defined on any finite additive group (and also on some other types of domains, though we will not discuss this point here). In particular, they can be defined on the finite-dimensional vector spaces {V} over a finite field {{\bf F}}.

In this case, the Gowers norms {U^{d+1}(V)} are closely tied to the space {\hbox{Poly}_{\leq d}(V \rightarrow {\bf R}/{\bf Z})} of polynomials of degree at most {d}. Indeed, as noted in Exercise 20 of Notes 4, a function {f: V \rightarrow {\bf C}} of {L^\infty(V)} norm {1} has {U^{d+1}(V)} norm equal to {1} if and only if {f = e(\phi)} for some {\phi \in \hbox{Poly}_{\leq d}(V \rightarrow {\bf R}/{\bf Z})}; thus polynomials solve the “{100\%} inverse problem” for the trivial inequality {\|f\|_{U^{d+1}(V)} \leq \|f\|_{L^\infty(V)}}. They are also a crucial component of the solution to the “{99\%} inverse problem” and “{1\%} inverse problem”. For the former, we will soon show:

Proposition 1 ({99\%} inverse theorem for {U^{d+1}(V)}) Let {f: V \rightarrow {\bf C}} be such that {\|f\|_{L^\infty(V)}} and {\|f\|_{U^{d+1}(V)} \geq 1-\epsilon} for some {\epsilon > 0}. Then there exists {\phi \in \hbox{Poly}_{\leq d}(V \rightarrow {\bf R}/{\bf Z})} such that {\| f - e(\phi)\|_{L^1(V)} = O_{d, {\bf F}}( \epsilon^c )}, where {c = c_d > 0} is a constant depending only on {d}.

Thus, for the Gowers norm to be almost completely saturated, one must be very close to a polynomial. The converse assertion is easily established:

Exercise 1 (Converse to {99\%} inverse theorem for {U^{d+1}(V)}) If {\|f\|_{L^\infty(V)} \leq 1} and {\|f-e(\phi)\|_{L^1(V)} \leq \epsilon} for some {\phi \in \hbox{Poly}_{\leq d}(V \rightarrow {\bf R}/{\bf Z})}, then {\|F\|_{U^{d+1}(V)} \geq 1 - O_{d,{\bf F}}( \epsilon^c )}, where {c = c_d > 0} is a constant depending only on {d}.

In the {1\%} world, one no longer expects to be close to a polynomial. Instead, one expects to correlate with a polynomial. Indeed, one has

Lemma 2 (Converse to the {1\%} inverse theorem for {U^{d+1}(V)}) If {f: V \rightarrow {\bf C}} and {\phi \in \hbox{Poly}_{\leq d}(V \rightarrow {\bf R}/{\bf Z})} are such that {|\langle f, e(\phi) \rangle_{L^2(V)}| \geq \epsilon}, where {\langle f, g \rangle_{L^2(V)} := {\bf E}_{x \in G} f(x) \overline{g(x)}}, then {\|f\|_{U^{d+1}(V)} \geq \epsilon}.

Proof: From the definition of the {U^1} norm (equation (18) from Notes 3), the monotonicity of the Gowers norms (Exercise 19 of Notes 3), and the polynomial phase modulation invariance of the Gowers norms (Exercise 21 of Notes 3), one has

\displaystyle  |\langle f, e(\phi) \rangle| = \| f e(-\phi) \|_{U^1(V)}

\displaystyle  \leq \|f e(-\phi) \|_{U^{d+1}(V)}

\displaystyle  = \|f\|_{U^{d+1}(V)}

and the claim follows. \Box

In the high characteristic case {\hbox{char}({\bf F}) > d} at least, this can be reversed:

Theorem 3 ({1\%} inverse theorem for {U^{d+1}(V)}) Suppose that {\hbox{char}({\bf F}) > d \geq 0}. If {f: V \rightarrow {\bf C}} is such that {\|f\|_{L^\infty(V)} \leq 1} and {\|f\|_{U^{d+1}(V)} \geq \epsilon}, then there exists {\phi \in \hbox{Poly}_{\leq d}(V \rightarrow {\bf R}/{\bf Z})} such that {|\langle f, e(\phi) \rangle_{L^2(V)}| \gg_{\epsilon,d,{\bf F}} 1}.

This result is sometimes referred to as the inverse conjecture for the Gowers norm (in high, but bounded, characteristic). For small {d}, the claim is easy:

Exercise 2 Verify the cases {d=0,1} of this theorem. (Hint: to verify the {d=1} case, use the Fourier-analytic identities {\|f\|_{U^2(V)} = (\sum_{\xi \in \hat V} |\hat f(\xi)|^4)^{1/4}} and {\|f\|_{L^2(V)} = (\sum_{\xi \in \hat V} |\hat f(\xi)|^2)^{1/2}}, where {\hat V} is the space of all homomorphisms {\xi: x \mapsto \xi \cdot x} from {V} to {{\bf R}/{\bf Z}}, and {\hat f(\xi) := \mathop{\bf E}_{x \in V} f(x) e(-\xi \cdot x)} are the Fourier coefficients of {f}.)

This conjecture for larger values of {d} are more difficult to establish. The {d=2} case of the theorem was established by Ben Green and myself in the high characteristic case {\hbox{char}({\bf F}) > 2}; the low characteristic case {\hbox{char}({\bf F}) = d = 2} was independently and simultaneously established by Samorodnitsky. The cases {d>2} in the high characteristic case was established in two stages, firstly using a modification of the Furstenberg correspondence principle, due to Ziegler and myself. to convert the problem to an ergodic theory counterpart, and then using a modification of the methods of Host-Kra and Ziegler to solve that counterpart, as done in this paper of Bergelson, Ziegler, and myself.

The situation with the low characteristic case in general is still unclear. In the high characteristic case, we saw from Notes 4 that one could replace the space of non-classical polynomials {\hbox{Poly}_{\leq d}(V \rightarrow {\bf R}/{\bf Z})} in the above conjecture with the essentially equivalent space of classical polynomials {\hbox{Poly}_{\leq d}(V \rightarrow {\bf F})}. However, as we shall see below, this turns out not to be the case in certain low characteristic cases (a fact first observed by Lovett, Meshulam, and Samorodnitsky, and independently by Ben Green and myself), for instance if {\hbox{char}({\bf F}) = 2} and {d \geq 3}; this is ultimately due to the existence in those cases of non-classical polynomials which exhibit no significant correlation with classical polynomials of equal or lesser degree. This distinction between classical and non-classical polynomials appears to be a rather non-trivial obstruction to understanding the low characteristic setting; it may be necessary to obtain a more complete theory of non-classical polynomials in order to fully settle this issue.

The inverse conjecture has a number of consequences. For instance, it can be used to establish the analogue of Szemerédi’s theorem in this setting:

Theorem 4 (Szemerédi’s theorem for finite fields) Let {{\bf F} = {\bf F}_p} be a finite field, let {\delta > 0}, and let {A \subset {\bf F}^n} be such that {|A| \geq \delta |{\bf F}^n|}. If {n} is sufficiently large depending on {p,\delta}, then {A} contains an (affine) line {\{ x, x+r, \ldots, x+(p-1)r\}} for some {x,r \in {\bf F}^n} with { r\neq 0}.

Exercise 3 Use Theorem 4 to establish the following generalisation: with the notation as above, if {k \geq 1} and {n} is sufficiently large depending on {p,\delta}, then {A} contains an affine {k}-dimensional subspace.

We will prove this theorem in two different ways, one using a density increment method, and the other using an energy increment method. We discuss some other applications below the fold.

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