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I have just uploaded to the arXiv the paper “An inverse theorem for an inequality of Kneser“, submitted to a special issue of the Proceedings of the Steklov Institute of Mathematics in honour of Sergei Konyagin. It concerns an inequality of Kneser discussed previously in this blog, namely that

\displaystyle \mu(A+B) \geq \min(\mu(A)+\mu(B), 1) \ \ \ \ \ (1)

whenever {A,B} are compact non-empty subsets of a compact connected additive group {G} with probability Haar measure {\mu}.  (A later result of Kemperman extended this inequality to the nonabelian case.) This inequality is non-trivial in the regime

\displaystyle \mu(A), \mu(B), 1- \mu(A)-\mu(B) > 0. \ \ \ \ \ (2)

The connectedness of {G} is essential, otherwise one could form counterexamples involving proper subgroups of {G} of positive measure. In the blog post, I indicated how this inequality (together with a more “robust” strengthening of it) could be deduced from submodularity inequalities such as

\displaystyle \mu( (A_1 \cup A_2) + B) + \mu( (A_1 \cap A_2) + B) \leq \mu(A_1+B) + \mu(A_2+B) \ \ \ \ \ (3)

which in turn easily follows from the identity {(A_1 \cup A_2) + B = (A_1+B) \cup (A_2+B)} and the inclusion {(A_1 \cap A_2) + B \subset (A_1 +B) \cap (A_2+B)}, combined with the inclusion-exclusion formula.

In the non-trivial regime (2), equality can be attained in (1), for instance by taking {G} to be the unit circle {G = {\bf R}/{\bf Z}} and {A,B} to be arcs in that circle (obeying (2)). A bit more generally, if {G} is an arbitrary connected compact abelian group and {\xi: G \rightarrow {\bf R}/{\bf Z}} is a non-trivial character (i.e., a continuous homomorphism), then {\xi} must be surjective (as {{\bf R}/{\bf Z}} has no non-trivial connected subgroups), and one can take {A = \xi^{-1}(I)} and {B = \xi^{-1}(J)} for some arcs {I,J} in that circle (again choosing the measures of these arcs to obey (2)). The main result of this paper is an inverse theorem that asserts that this is the only way in which equality can occur in (1) (assuming (2)); furthermore, if (1) is close to being satisfied with equality and (2) holds, then {A,B} must be close (in measure) to an example of the above form {A = \xi^{-1}(I), B = \xi^{-1}(J)}. Actually, for technical reasons (and for the applications we have in mind), it is important to establish an inverse theorem not just for (1), but for the more robust version mentioned earlier (in which the sumset {A+B} is replaced by the partial sumset {A +_\varepsilon B} consisting of “popular” sums).

Roughly speaking, the idea is as follows. Let us informally call {(A,B)} a critical pair if (2) holds and the inequality (1) (or more precisely, a robust version of this inequality) is almost obeyed with equality. The notion of a critical pair obeys some useful closure properties. Firstly, it is symmetric in {A,B}, and invariant with respect to translation of either {A} or {B}. Furthermore, from the submodularity inequality (3), one can show that if {(A_1,B)} and {(A_2,B)} are critical pairs (with {\mu(A_1 \cap A_2)} and {1 - \mu(A_1 \cup A_2) - \mu(B)} positive), then {(A_1 \cap A_2,B)} and {(A_1 \cup A_2, B)} are also critical pairs. (Note that this is consistent with the claim that critical pairs only occur when {A,B} come from arcs of a circle.) Similarly, from associativity {(A+B)+C = A+(B+C)}, one can show that if {(A,B)} and {(A+B,C)} are critical pairs, then so are {(B,C)} and {(A,B+C)}.

One can combine these closure properties to obtain further ones. For instance, suppose {A,B} is such that {\mu(A+B) < 2 \mu(A)}, so that all translates {A+b}, {b \in B} intersect each other in a set of positive measure. Suppose also that {(A,C)} is a critical pair and {1-\mu(A+B) - \mu(C) > 0}. Then (cheating a little bit), one can show that {(A+B,C)} is also a critical pair, basically because {A+B} is the union of the {A+b}, {b \in B}, the {(A+b,C)} are all critical pairs, and the {A+b} all intersect each other. This argument doesn’t quite work as stated because one has to apply the closure property under union an uncountable number of times, but it turns out that if one works with the robust version of sumsets and uses a random sampling argument to approximate {A+B} by the union of finitely many of the {A+b}, then the argument can be made to work.

Using all of these closure properties, it turns out that one can start with an arbitrary critical pair {(A,B)} and end up with a small set {C} such that {(A,C)} and {(kC,C)} are also critical pairs for all {1 \leq k \leq 10^4} (say), where {kC} is the {k}-fold sumset of {C}. (Intuitively, if {A,B} are thought of as secretly coming from the pullback of arcs {I,J} by some character {\xi}, then {C} should be the pullback of a much shorter arc by the same character.) In particular, {C} exhibits linear growth, in that {\mu(kC) = k\mu(C)} for all {1 \leq k \leq 10^4}. One can now use standard technology from inverse sumset theory to show first that {C} has a very large Fourier coefficient (and thus is biased with respect to some character {\xi}), and secondly that {C} is in fact almost of the form {C = \xi^{-1}(K)} for some arc {K}, from which it is not difficult to conclude similar statements for {A} and {B} and thus finish the proof of the inverse theorem.

In order to make the above argument rigorous, one has to be more precise about what the modifier “almost” means in the definition of a critical pair. I chose to do this in the language of “cheap” nonstandard analysis (aka asymptotic analysis), as discussed in this previous blog post; one could also have used the full-strength version of nonstandard analysis, but this does not seem to convey any substantial advantages. (One can also work in a more traditional “non-asymptotic” framework, but this requires one to keep much more careful account of various small error terms and leads to a messier argument.)


[Update, Nov 15: Corrected the attribution of the inequality (1) to Kneser instead of Kemperman.  Thanks to John Griesmer for pointing out the error.]

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|


\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.

For a number of reasons, including the start of the summer break for me and my coauthors, a number of papers that we have been working on are being released this week.  For instance, Ben Green and I have just uploaded to the arXiv our paper “An equivalence between inverse sumset theorems and inverse conjectures for the U^3 norm“, submitted to Math. Proc. Camb. Phil. Soc..  The main result of this short paper (which was briefly announced in this earlier post) is a connection between two types of inverse theorems in additive combinatorics, namely the inverse sumset theorems of Freiman type, and inverse theorems for the Gowers uniformity norm, and more specifically, for the U^3 norm

\|f\|_{U^3(G)}^8 := {\Bbb E}_{x,a,b,c \in G} f(x) \overline{f(x+a)} \overline{f(x+b)} \overline{f(x+c)} f(x+a+b) f(x+a+c) f(x+b+c) \overline{f(x+a+b+c)}

on finite additive group G, where f: G \to {\Bbb C} is a complex-valued function.

As usual, the connection is easiest to state in a finite field model such as G = {\Bbb F}_2^n.  In this case, we have the following inverse sumset theorem of Ruzsa:

Theorem 1. If A \subset {\Bbb F}_2^n is such that |A+A| \leq K|A|, then A can be covered by a translate of a subspace of {\Bbb F}_2^n of cardinality at most K^2 2^{K^4} |A|.

The constant K^2 2^{K^4} has been improved for large K in a sequence of papers, from K 2^{\lfloor K^3 \rfloor-1} by Ruzsa, K^2 2^{K^2-2} by Green-Ruzsa, 2^{O(K^{3/2} \log(1+K)} by Sanders, 2^{2K+O(\sqrt{K} \log K}) by Green and myself, and finally 2^{2K+O(\log K)} by Konyagin (private communication) which is sharp except for the precise value of the O() implied constant (as can be seen by considering the example when A consists of about 2K independent elements).  However, it is conjectured that the polynomial loss can be removed entirely if one modifies the conclusion slightly:

Conjecture 1. (Polynomial Freiman-Ruzsa conjecture for {\Bbb F}_2^n.) If A \subset {\Bbb F}_2^n is such that |A+A| \leq K|A|, then A can be covered by O(K^{O(1)}) translates of subspaces of {\Bbb F}_2^n of cardinality at most |A|.

This conjecture was verified for downsets by Green and myself, but is open in general.   This conjecture has a number of equivalent formulations; see this paper of Green for more discussion.  In this previous post we show that a stronger version of this conjecture fails.

Meanwhile, for the Gowers norm, we have the following inverse theorem, due to Samorodnitsky:

Theorem 2. Let f: {\Bbb F}_2^n \to [-1,1] be a function whose U^3 norm is at least 1/K.  Then there exists a quadratic polynomial Q: {\Bbb F}_2^n \to {\Bbb F}_2 such that |{\Bbb E}_{x \in {\Bbb F}_2^n} f(x) (-1)^{Q(x)}| \geq \exp( - O(K)^{O(1)} ).

Note that the quadratic phases (-1)^{Q(x)} are the only functions taking values in [-1,1] whose U^3 norm attains its maximal value of 1.

It is conjectured that the exponentially weak correlation here can be strengthened to a polynomial one:

Conjecture 2. (Polynomial inverse conjecture for the U^3({\Bbb F}_2^n) norm). Let f: {\Bbb F}_2^n \to [-1,1] be a function whose U^3 norm is at least 1/K.  Then there exists a quadratic polynomial Q: {\Bbb F}_2^n \to {\Bbb F}_2 such that |{\Bbb E}_{x \in {\Bbb F}_2^n} f(x) (-1)^{Q(x)}| \geq K^{-O(1)}.

The first main result of this paper is

Theorem 3. Conjecture 1 and Conjecture 2 are equivalent.

This result was also independently observed by Shachar Lovett (private communication).  We also establish an analogous result for the cyclic group {\Bbb Z}/N{\Bbb Z}, in which the notion of polynomial is replaced by that of a subexponential \exp(K^{o(1)}), and in which the notion of a quadratic polynomial is replaced by a 2-step nilsequence; the precise statement is a bit technical and will not be given here.  We also observe a partial partial analogue of the correpsondence between inverse sumset theorems and Gowers norms in the higher order case, in particular observing that U^4 inverse theorems imply a certain rigidity result for “Freiman-quadratic polynomials” (a quadratic version of Conjecture 3 below).

Below the fold, we sketch the proof of Theorem 3.

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I’ve just uploaded to the arXiv my paper “An inverse theorem for the bilinear $L^2$ Strichartz estimate for the wave equation“.  This paper is another technical component of my “heatwave project“, which aims to establish the global regularity conjecture for energy-critical wave maps into hyperbolic space.    I have been in the process of writing the final paper of that project, in which I will show that the only way singularities can form is if a special type of solution, known as an “almost periodic blowup solution”, exists.  However, I recently discovered that the existing function space estimates that I was relying on for the large energy perturbation theory were not quite adequate, and in particular I needed a certain “inverse theorem” for a standard bilinear estimate which was not quite in the literature.  The purpose of this paper is to establish that inverse theorem, which may also have some application to other nonlinear wave equations.

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