Tanja Eisner and I have just uploaded to the arXiv our paper “Large values of the Gowers-Host-Kra seminorms“, submitted to Journal d’Analyse Mathematique. This paper is concerned with the properties of three closely related families of (semi)norms, indexed by a positive integer {k}:

  • The Gowers uniformity norms {\|f\|_{U^k(G)}} of a (bounded, measurable, compactly supported) function {f: G \rightarrow {\bf C}} taking values on a locally compact abelian group {G}, equipped with a Haar measure {\mu};
  • The Gowers uniformity norms {\|f\|_{U^k([N])}} of a function {f: [N] \rightarrow {\bf C}} on a discrete interval {\{1,\ldots,N\}}; and
  • The Gowers-Host-Kra seminorms {\|f\|_{U^k(X)}} of a function {f \in L^\infty(X)} on an ergodic measure-preserving system {X = (X,{\mathcal X},\mu,T)}.

These norms have been discussed in depth in previous blog posts, so I will just quickly review the definition of the first norm here (the other two (semi)norms are defined similarly). The {U^k(G)} norm is defined recursively by setting

\displaystyle \| f \|_{U^1(G)} := |\int_G f\ d\mu|

and

\displaystyle \|f\|_{U^k(G)}^{2^k} := \int_G \| \Delta_h f \|_{U^{k-1}(G)}^{2^{k-1}}\ d\mu(h)

where {\Delta_h f(x) := f(x+h) \overline{f(x)}}. Equivalently, one has

\displaystyle \|f\|_{U^k(G)} := (\int_G \ldots \int_G \Delta_{h_1} \ldots \Delta_{h_k} f(x)\ d\mu(x) d\mu(h_1) \ldots d\mu(h_k))^{1/2^k}.

Informally, the Gowers uniformity norm {\|f\|_{U^k(G)}} measures the extent to which (the phase of {f}) behaves like a polynomial of degree less than {k}. Indeed, if {\|f\|_{L^\infty(G)} \leq 1} and {G} is compact with normalised Haar measure {\mu(G)=1}, it is not difficult to show that {\|f\|_{U^k(G)}} is at most {1}, with equality if and only if {f} takes the form {f = e(P) := e^{2\pi iP}} almost everywhere, where {P: G \rightarrow {\bf R}/{\bf Z}} is a polynomial of degree less than {k} (which means that {\partial_{h_1} \ldots \partial_{h_k} P(x) = 0} for all {x,h_1,\ldots,h_k \in G}).

Our first result is to show that this result is robust, uniformly over all choices of group {G}:

Theorem 1 ({L^\infty}-near extremisers) Let {G} be a compact abelian group with normalised Haar measure {\mu(G)=1}, and let {f \in L^\infty(G)} be such that {\|f\|_{L^\infty(G)} \leq 1} and {\|f\|_{U^k(G)} \geq 1-\epsilon} for some {\epsilon > 0} and {k \geq 1}. Then there exists a polynomial {P: G \rightarrow {\bf R}/{\bf Z}} of degree at most {k-1} such that {\|f-e(P)\|_{L^1(G)} = o(1)}, where {o(1)} is bounded by a quantity {c_k(\epsilon)} that goes to zero as {\epsilon \rightarrow 0} for fixed {k}.

The quantity {o(1)} can be described effectively (it is of polynomial size in {\epsilon}), but we did not seek to optimise it here. This result was already known in the case of vector spaces {G = {\bf F}_p^n} over a fixed finite field {{\bf F}_p} (where it is essentially equivalent to the assertion that the property of being a polynomial of degree at most {k-1} is locally testable); the extension to general groups {G} turns out to fairly routine. The basic idea is to use the recursive structure of the Gowers norms, which tells us in particular that if {\|f\|_{U^k(G)}} is close to one, then {\|\Delta_h f\|_{U^{k-1}(G)}} is close to one for most {h}, which by induction implies that {\Delta_h f} is close to {e(Q_h)} for some polynomials {Q_h} of degree at most {k-2} and for most {h}. (Actually, it is not difficult to use cocycle equations such as {\Delta_{h+k} f = \Delta_h f \times T^h \Delta_k f} (when {|f|=1}) to upgrade “for most {h}” to “for all {h}“.) To finish the job, one would like to express the {Q_h} as derivatives {Q_h = \partial_h P} of a polynomial {P} of degree at most {k-1}. This turns out to be equivalent to requiring that the {Q_h} obey the cocycle equation

\displaystyle Q_{h+k} = Q_h + T^h Q_k

where {T^h F(x) := F(x+h)} is the translate of {F} by {h}. (In the paper, the sign conventions are reversed, so that {T^h F(x) := F(x-h)}, in order to be compatible with ergodic theory notation, but this makes no substantial difference to the arguments or results.) However, one does not quite get this right away; instead, by using some separation properties of polynomials, one can show the weaker statement that

\displaystyle Q_{h+k} = Q_h + T^h Q_k + c_{h,k} \ \ \ \ \ (1)

where the {c_{h,k}} are small real constants. To eliminate these constants, one exploits the trivial cohomology of the real line. From (1) one soon concludes that the {c_{h,k}} obey the {2}-cocycle equation

\displaystyle c_{h,k} + c_{h+k,l} = c_{h,k+l} + c_{k,l}

and an averaging argument then shows that {c_{h,k}} is a {2}-coboundary in the sense that

\displaystyle c_{h,k} = b_{h+k} - b_h - b_k

for some small scalar {b_h} depending on {h}. Subtracting {b_h} from {Q_h} then gives the claim.

Similar results and arguments also hold for the {U^k([N])} and {U^k(X)} norms, which we will not detail here.

Dimensional analysis reveals that the {L^\infty} norm is not actually the most natural norm with which to compare the {U^k} norms against. An application of Young’s convolution inequality in fact reveals that one has the inequality

\displaystyle \|f\|_{U^k(G)} \leq \|f\|_{L^{p_k}(G)} \ \ \ \ \ (2)

where {p_k} is the critical exponent {p_k := 2^k/(k+1)}, without any compactness or normalisation hypothesis on the group {G} and the Haar measure {\mu}. This allows us to extend the {U^k(G)} norm to all of {L^{p_k}(G)}. There is then a stronger inverse theorem available:

Theorem 2 ({L^{p_k}}-near extremisers) Let {G} be a locally compact abelian group, and let {f \in L^{p_k}(G)} be such that {\|f\|_{L^{p_k}(G)} \leq 1} and {\|f\|_{U^k(G)} \geq 1-\epsilon} for some {\epsilon > 0} and {k \geq 1}. Then there exists a coset {H} of a compact open subgroup {H} of {G}, and a polynomial {P: H to {\bf R}/{\bf Z}} of degree at most {k-1} such that {\|f-e(P) 1_H\|_{L^{p_k}(G)} = o(1)}.

Conversely, it is not difficult to show that equality in (2) is attained when {f} takes the form {e(P) 1_H} as above. The main idea of proof is to use an inverse theorem for Young’s inequality due to Fournier to reduce matters to the {L^\infty} case that was already established. An analogous result is also obtained for the {U^k(X)} norm on an ergodic system; but for technical reasons, the methods do not seem to apply easily to the {U^k([N])} norm. (This norm is essentially equivalent to the {U^k({\bf Z}/\tilde N{\bf Z})} norm up to constants, with {\tilde N} comparable to {N}, but when working with near-extremisers, norms that are only equivalent up to constants can have quite different near-extremal behaviour.)

In the case when {G} is a Euclidean group {{\bf R}^d}, it is possible to use the sharp Young inequality of Beckner and of Brascamp-Lieb to improve (2) somewhat. For instance, when {k=3}, one has

\displaystyle \|f\|_{U^3({\bf R}^d)} \leq 2^{-d/8} \|f\|_{L^2({\bf R}^d)}

with equality attained if and only if {f} is a gaussian modulated by a quadratic polynomial phase. This additional gain of {2^{-d/8}} allows one to pinpoint the threshold {1-\epsilon} for the previous near-extremiser results in the case of {U^3} norms. For instance, by using the Host-Kra machinery of characteristic factors for the {U^3(X)} norm, combined with an explicit and concrete analysis of the {2}-step nilsystems generated by that machinery, we can show that

\displaystyle \|f\|_{U^3(X)} \leq 2^{-1/8} \|f\|_{L^2(X)}

whenever {X} is a totally ergodic system and {f} is orthogonal to all linear and quadratic eigenfunctions (which would otherwise form immediate counterexamples to the above inequality), with the factor {2^{-1/8}} being best possible. We can also establish analogous results for the {U^3([N])} and {U^3({\bf Z}/N{\bf Z})} norms (using the inverse {U^3} theorem of Ben Green and myself, in place of the Host-Kra machinery), although it is not clear to us whether the {2^{-1/8}} threshold remains best possible in this case.