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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.

A (complex, semi-definite) inner product space is a complex vector space ${V}$ equipped with a sesquilinear form ${\langle, \rangle: V \times V \rightarrow {\bf C}}$ which is conjugate symmetric, in the sense that ${\langle w, v \rangle = \overline{\langle v, w \rangle}}$ for all ${v,w \in V}$, and non-negative in the sense that ${\langle v, v \rangle \geq 0}$ for all ${v \in V}$. By inspecting the non-negativity of ${\langle v+\lambda w, v+\lambda w\rangle}$ for complex numbers ${\lambda \in {\bf C}}$, one obtains the Cauchy-Schwarz inequality

$\displaystyle |\langle v, w \rangle| \leq |\langle v, v \rangle|^{1/2} |\langle w, w \rangle|^{1/2};$

if one then defines ${\|v\| := |\langle v, v \rangle|^{1/2}}$, one then quickly concludes the triangle inequality

$\displaystyle \|v + w \| \leq \|v\| + \|w\|$

which then soon implies that ${\| \|}$ is a semi-norm on ${V}$. If we make the additional assumption that the inner product ${\langle,\rangle}$ is positive definite, i.e. that ${\langle v, v \rangle > 0}$ whenever ${v}$ is non-zero, then this semi-norm becomes a norm. If ${V}$ is complete with respect to the metric ${d(v,w) := \|v-w\|}$ induced by this norm, then ${V}$ is called a Hilbert space.

The above material is extremely standard, and can be found in any graduate real analysis course; I myself covered it here. But what is perhaps less well known (except inside the fields of additive combinatorics and ergodic theory) is that the above theory of classical Hilbert spaces is just the first case of a hierarchy of higher order Hilbert spaces, in which the binary inner product ${f, g \mapsto \langle f, g \rangle}$ is replaced with a ${2^d}$-ary inner product ${(f_\omega)_{\omega \in \{0,1\}^d} \mapsto \langle (f_\omega)_{\omega \in \{0,1\}^d}}$ that obeys an appropriate generalisation of the conjugate symmetry, sesquilinearity, and positive semi-definiteness axioms. Such inner products then obey a higher order Cauchy-Schwarz inequality, known as the Cauchy-Schwarz-Gowers inequality, and then also obey a triangle inequality and become semi-norms (or norms, if the inner product was non-degenerate). Examples of such norms and spaces include the Gowers uniformity norms ${\| \|_{U^d(G)}}$, the Gowers box norms ${\| \|_{\Box^d(X_1 \times \ldots \times X_d)}}$, and the Gowers-Host-Kra seminorms ${\| \|_{U^d(X)}}$; a more elementary example are the family of Lebesgue spaces ${L^{2^d}(X)}$ when the exponent is a power of two. They play a central role in modern additive combinatorics and to certain aspects of ergodic theory, particularly those relating to Szemerédi’s theorem (or its ergodic counterpart, the Furstenberg multiple recurrence theorem); they also arise in the regularity theory of hypergraphs (which is not unrelated to the other two topics).

A simple example to keep in mind here is the order two Hilbert space ${L^4(X)}$ on a measure space ${X = (X,{\mathcal B},\mu)}$, where the inner product takes the form

$\displaystyle \langle f_{00}, f_{01}, f_{10}, f_{11} \rangle_{L^4(X)} := \int_X f_{00}(x) \overline{f_{01}(x)} \overline{f_{10}(x)} f_{11}(x)\ d\mu(x).$

In this brief note I would like to set out the abstract theory of such higher order Hilbert spaces. This is not new material, being already implicit in the breakthrough papers of Gowers and Host-Kra, but I just wanted to emphasise the fact that the material is abstract, and is not particularly tied to any explicit choice of norm so long as a certain axiom are satisfied. (Also, I wanted to write things down so that I would not have to reconstruct this formalism again in the future.) Unfortunately, the notation is quite heavy and the abstract axiom is a little strange; it may be that there is a better way to formulate things. In this particular case it does seem that a concrete approach is significantly clearer, but abstraction is at least possible.

Note: the discussion below is likely to be comprehensible only to readers who already have some exposure to the Gowers norms.