On a conjecture of Gowers and Long

We show that rounding to a delta-net in SO(3) is not close to a group operation, thus confirming a conjecture of Gowers and Long.


I
In a very interesting recent preprint [7], Gowers and Long considered somewhat associative binary operations. In their paper they describe a very natural example of a somewhat associative operation, and conjecture (see [7,Conjecture 1.6]) that it does not resemble a genuine group operation. Our aim in this note is to prove their conjecture. Throughout the paper we will take δ > 0 be a small parameter, and let X be a maximal δ-separated subset of SO (3). We have |X| ∼ δ −3 . Define a binary operation • : X ×X → X by defining x • y to be the nearest point of X to xy (breaking ties arbitrarily). Since X is assumed to be maximal δ-separated, we always have (1.1) d(x • y, xy) δ.
Claim (Gowers-Long). For a positive proportion of triples (x 1 , x 2 , x 3 ) ∈ X 3 we have the associativity relation ( Gowers and Long note that it seems very unlikely that any substantial portion of the multiplication table of • can be embedded in a group operation, and make a precise conjecture, [7,Conjecture 1.6], to this effect. The following result establishes their conjecture. Theorem 1.1. Suppose that ι : X → G is an injective map into a group G, with group operation ·. Then the number of pairs (x 1 , x 2 ) ∈ X × X with ι(x 1 ) · ι(x 2 ) = ι(x 1 • x 2 ) is at most ε|X| 2 , where ε → 0 as δ → 0.
We remark that we do not obtain any effective information on the speed at which ε → 0. This is because we rely on the structure theory of approximate groups [2], which uses ultrafilter arguments.
Notation. Our notation is fairly standard. Occasionally we will write, for instance, o K;δ→0 (1), which means some quantity tending to zero as δ → 0, but the rate at which 2000 Mathematics Subject Classification. Primary . The author is supported by a Simons Investigator Grant. this happens may depend on the parameter K. We write X ≫ Y to mean X cY for some absolute c > 0, and X ∼ Y means Y ≪ X ≪ Y .
Recall that · denotes the Hilbert-Schmidt norm and that we use this to define a distance on SO(3) by d(g, h) := g − h , where SO(3) is embedded in the space of 3-by-3 matrices by fixing an orthonormal basis for R 3 . We write |g| = d(g, 1) for the distance to the identity (which we will always denote by 1, the underlying group hopefully being clear from context). It may be computed that |g| = 2 3/2 | sin(θ/2)|, where θ is the angle of the rotation g. Recall that the Hilbert-Schmidt norm is submultiplicative (that is, ab a b for all real 3-by-3 matrices). Additionally, using the conjugation invariance of trace and the fact that g T = g −1 for g ∈ SO(3), we have the SO(3)invariance a = ag = ga for all 3-by-3 matrices a and all g ∈ SO(3). In particular, Acknowledgement. I would like to thank Emmanuel Breuillard for helpful conversations.

A
We can fairly quickly reduce the task of proving Theorem 1.1 to that of establishing the following proposition which, since it does not involve the awkward •, is of a more conventional type. Proposition 2.1. Let ε, δ > 0. Let (G, ·) be a group, and let A be a finite subset of G of size n. Let f : A → SO(3) be a map with δ-separated image, and with the property that there are at least εn 3 quadruples (a 1 , a 2 , a 3 , a 4 ) ∈ A 4 with a 1 · a 2 = a 3 · a 4 and d(f (a 1 )f (a 2 ), f (a 3 )f (a 4 )) δ. Then n = o ε;δ→0 (δ −3 ).
It follows from the triangle inequality that Applying Proposition 2.1 (with δ replaced by 2δ and ε by ε 2 ), we see that |X| = n = o ε;δ→0 (δ −3 ), a contradiction if δ is small enough as a function of ε.

O
In this section we outline the rest of the argument. Recall that a K-approximate group is a subset B of some ambient group which is symmetric (that is, it contains the identity 1, and B −1 = B) and such that B 2 is covered by K left-(or equivalently right-) translates of B. See [11] for further discussion and background. Note that approxmate groups need not be finite.
In the next section, we show that the existence of a map f as in Proposition 2.1 would imply the existence of an approximate homomorphism from a finite approximate group to SO(3) with a "thick" image. In discussing approximate homomorphisms φ it is natural to introduce the notion of cocyle, defining ∂φ(x, y) := φ(y) −1 φ(x) −1 φ(xy).
Proposition 3.1. Let ε, δ > 0. Let (G, ·) be a group, and let A be a finite subset of G of size n ∼ δ −3 . Let f : A → SO(3) be a map with δ-separated image, and with the property that there are at least εn 3 quadruples (a 1 , a 2 , a 3 , a 4 ) ∈ A 4 with a 1 · a 2 = a 3 · a 4 and Remarks. Here N δ means the δ-neighbourhood (in the metric d) and µ is the normalised Haar measure on SO(3). As we shall see, no particular properties of SO(3) are used in the proof, beyond the existence of d and µ and their basic properties. Note that the "approximateness" of φ, whilst of two different types (the error set S and the parameter δ) is all in the range, whereas f is approximate in the domain, in that the weak homomorphism property only holds some of the time. This idea of moving the ambiguity from the domain to the range follows a line of argument pioneered by Gowers in his seminal works [5,6] (based also on work of Ruzsa). Proposition 3.1 is a consequence of the metric entropy version of the noncommutative Balog-Szemerédi-Gowers theorem of Tao [11]. Proposition 2.1, and hence Theorem 1.1, follows immediately from Proposition 3.1 and the next result, which says that the two properties of φ in the conclusion of Proposition 3.1 are incompatible: an approximate homomorphism from a finite approximate group to SO(3) has a thin image.
Proposition 3.2. Let B be a finite K-approximate group, let S ⊂ SO(3) be a set of size at most K, and suppose that φ : The proof of this uses quite different techniques and we appeal to some fairly specific features of SO(3), though it would probably be possible to adapt the proof so as to work with SO(3) replaced by, for example, any simple Lie group. We divide into two cases, according to whether or not the cocycle ∂φ takes values far from the identity. Recall that, for g ∈ SO(3), |g| means d(g, 1).
Then a fairly direct argument shows that φ(B 4 ) must lie in a union of O(1) translates of an "almost centraliser" of ∂φ(x, y), and a further argument shows this has small measure.
were actually a finite group, a result of Kazhdan [9] then implies that we can correct φ by O( √ δ) to get a genuine homomorphismφ : However, these are all cyclic, dihedral or contained in S 5 and hence φ(B 4 ) is "thin" in the sense discussed above. It is tempting to try and minic the arguments of [9] when B is merely an approximate group, but this does not work in any obvious way. Rather we use the classification of approximate groups due to Breuillard, Tao and the author [2] and then invoke Kazhdan's result as a black box, ending up showing that φ(B 4 ) can be partitioned into O(1) pieces, each of which almost satisfies a nontrivial word equation, which then implies that φ(B 4 ) is thin. In particular we do not prove that φ can be corrected to a homomorphismφ; it would be interesting to explore this direction. For further remarks see Section 7.

A T ' BSG
In this section we establish Proposition 3.1 . Let G := G×SO (3), and let d : G×G → G be the product of the discrete (extended) metric d triv on G, where the distance between distinct points is ∞, and the metric d on SO(3). Let µ = µ triv × µ, where µ triv is the counting measure on G (that is, the measure of any finite set A ⊂ G is simply |A|) and µ is normalised Haar measure on SO(3). The group G, endowed with the measure µ and the (extended) metric d, is locally reasonable in the sense of Tao [11, Definition 6.3] 1 .
To state the result from [11] that we will need, we recall the definition of covering numbers used in that paper: if X is a subset of a metric space, N η (X) is the least number of balls of radius η necessary to cover X. We also define the η-approximate multiplicative energy E η (X, X) of a set to be N η (Q η (X, X)), where The metric entropy here is with respect to the product metric on X 4 .
The following is the implication (i) ⇒ (iv) of [11, Theorem 6.10], specialised to our setting.  (a 1 , a 2 , a 3 , a 4 ) with a 1 · a 2 = a 3 · a 4 and d(f (a 1 )f (a 2 ), f (a 3 )f (a 4 )) δ. , f (a)) : a ∈ A} ⊂ G to be the graph of f . Let π : G → G be projection. Then, since π is injective on X and the metric on G is discrete, By assumption, |Q δ (X, X)| εn 3 . Since π ⊗4 : G 4 → G 4 is injective on Q δ (X, X), we have E δ (X, X) = |Q δ (X, X)| εn 3 . Therefore the hypothesis of Proposition 4.1 is satisfied with K = ε −1 . For the rest of the proof of Proposition 3.1, all instances of the ≫, ≪ and O() notations may depend on ε but this will not be explicitly indicated. Applying Proposition 4.1, we obtain an O(1)-approximate subgroup H ⊂ G and an element g = (x, y) ∈ G satisfying Let B 0 := π(g −1 X ∩ H). Using the fact that π is injective on X and that G is discrete, we have Using the fact that the metric on G is discrete once more, we also have |π(H)| N δ (H), and hence from (4.1), (4.2) it follows that (Note that H itself may well be infinite). Let φ : Extend φ to a map from B 12 to SO(3) as follows: for each x ∈ B 12 \ B 0 , write x = b ε1 1 · · · b ε36 36 with ε 1 , . . . , ε 36 ∈ {−1, 0, 1} and b 1 , . . . , b 36 ∈ B 0 . If there is more than one such representation of a given x, choose one arbitrarily. Now define Note that φ(B 4 ) contains φ(B 0 ) which, as observed above, is a collection of ∼ n δseparated points. Since n ∼ δ −3 (and the volume of a (δ/2)-neighbourhood in SO(3) is ∼ δ 3 ), it follows that µ(N δ (φ(B 4 ))) ∼ 1.
To prove (4.4), observe first that, since H is a K-approximate group for some K = O(1), H 37 is covered by K 36 translates of H, and so (by (4.1) and the bi-invariance of the metric) we have (4.5) N δ (H 37 ) ∼ n.
However, H 37 contains a translate of F above every point of π(H), and thus The desired estimate (4.4) follows immediately from this, (4.3) and (4.5).

C 1:
We turn now to the proof of Proposition 3.2. The reader may wish to recall the outline given in Section 3. In this section we look at the first case discussed there, in which there are x, y ∈ B 4 such that |∂φ(x, y)| > √ δ. Before giving the main argument, let us record a lemma concerning almost commuting rotations. This must surely exist in the literature but I could not locate a reference. Here, and in what follows, we define the conjugate a g to be g −1 ag and the commutator [a, g] to be a −1 g −1 ag. In case (i), |r(α)| = 2 3/2 | sin(α/2)| = 2 3/2 y ≪ η/|a|. Therefore d(g, C(a)) d(g, r(β 1 + β 2 )) = r(β 1 )(r ′ (α) − 1)r(β 2 ) 8|r ′ (α)| ≪ η/|a|, where in the penultimate step we used the submultiplicativity of · . This concludes the proof in this case.
In case (ii), d(r ′ (α), r ′ (π)) = 2 3/2 | cos(α/2)| ≪ η. Noting that r ′ (π) commutes with a, we have This completes the proof of the lemma. Now we return to the proof of Proposition 3.2 (first case). Let z ∈ B 4 be arbitrary. By writing φ(xyz) in two different ways one easily obtains the cocycle equation where a := ∂φ(x, y). Since x, y, z ∈ B 4 , all the pairwise products as well as the triple product xyz lie in B 12 , and hence by the hypotheses of Proposition 3.2 the three cocycles ∂φ(y, z), ∂φ(x, yz), ∂φ(xy, z) lie in the δ-neighbourhood of S. It follows from (5.1) that, for all z ∈ B 4 , d(a φ(z) , SSS −1 ) 3δ. Consequently, we may find a set z 1 , . . . , z k , k |SSS −1 | K 3 , of elements of B 4 such that for every z ∈ B 4 there is some i such that d(a φ(z) , a φ(zi) ) 6δ. Equivalently, |[a, φ(z)φ(z i ) −1 ]| 6δ. By Lemma 5.1 we see that for every z ∈ B 4 there is some i such that d(φ(z)φ(z i ) −1 , C(a)) ≪ √ δ. Thus φ(B 4 ) is contained in the ( √ δ)-neighbourhood of at most k translates of C(a), a set whose measure tends to 0 as δ → 0, uniformly in a = 1. This concludes the proof in the first case.
We will repeatedly use the fact, easily established using (6.1) and induction, that if Q is a symmetric set with Q m ⊂ B 12 then for all x, y ∈ Q, where w is any word of length at most m in the variables x, y. Of particular interest to us will be the commutator words w 1 (a, b) := [a, b], w i+1 (a, b) := [a, w i (a, b)]. The length of w s is ℓ(w s ) = 3 · 2 s − 2.
We will also use the following result of Breuillard, Tao and the author [2]. , and with |P | ≫ K |B|. We refer the reader to [2, Section 2] for the definitions required here, though the reader can fairly happily treat these concepts as black boxes for the purpose of this discussion. In particular, since P contains H, so does B 4 . Set m = 10ℓ(w s ), thus 4 m ≪ K 1, and let Q := P H (u 1 , . . . , u r ; 1 m N 1 , . . . , 1 m N r ). It follows from the definitions in [2, Section 2] that Q is symmetric and Q m ⊂ P , and it follows from [2, Lemma C.1] that |Q| ≫ K,m |P |. Finally, if x, y ∈ Q 4 then certainly x, y ∈ P , and so from the fact that P/H is s-step nilpotent we see that indeed w s (x, y) ∈ H.
From now on we drop explicit mention of K; all bounds can (and will) depend on K. It follows from the (nonabelian) Ruzsa covering lemma [11,Lemma 3.6] that B 4 is a union of O(1) translates Q 2 g i , where g i ∈ B 4 . Evidently it suffices to show that µ(N δ (φ(Q 2 g i ))) = o δ→0 (1) for each i. Fix some i and set g := g i .

F
We have already remarked that it would be interesting to understand more about the structure of approximate homomorphisms φ : B → SO(3) where B is an approximate group. Does an analogue of Kazhdan's theorem hold for them, that is to say if (6.1) holds, 2 Kazhdan acknowledges that in the compact case the result was obtained earlier by Grove, Karcher and Ruh [8], and in fact similar ideas go back to Turing [12]. 3 One could get away with weaker results here, such as Jordan's theorem. 4 Any word map is Lipschitz with Lipschitz constant the length of the word, since if d(t i , t ′ i ) δ then d(t 1 · · · tm, t ′ 1 · · · t ′ m ) mδ, by an easy induction.
is thereφ : B → SO(3) satisfyingφ(xy) =φ(x)φ(y) and with d(φ(x),φ(x)) = O(δ) for all x? It might be possible to answer this question using the thesis of Carolino [3], applied to the graph of φ. This would allow for an alternative to the arguments of Section 6 by appealing to [1], which says that finite approximate subgroups of SO(3) are almost abelian.
The example of Gowers and Long considered in this paper is natural, but has the slightly unsatisfactory property that the operation • is not cancellative. It is only weakly cancellative in the sense that for a given x and z there are at most O(1) values of y for which x • y = z. I have some notes on a potential example which is fully cancellative, so its multiplication table is a latin square. Roughly speaking, it comes from replacing SO(3) by a compact portion of the Heisenberg group (Jason Long informs me that he and Gowers also considered such examples). I initially thought that the Heisenberg group, being almost abelian, would be much easier to analyse than SO (3), but this turned out not to be the case. The main reason is that the Heisenberg group does contain approximate subgroups with "thick" image.