Bounds in Cohen’s Idempotent Theorem

Suppose that G is a finite Abelian group and write W(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {W}}(G)$$\end{document} for the set of cosets of subgroups of G. We show that if f:G→Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:G \rightarrow {\mathbb {Z}}$$\end{document} satisfies the estimate ‖f‖A(G)≤M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert f\Vert _{A(G)} \le M$$\end{document} with respect to the Fourier algebra norm, then there is some z:W(G)→Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z:{\mathcal {W}}(G) \rightarrow {\mathbb {Z}}$$\end{document} such that f=∑W∈W(G)z(W)1Wand‖z‖ℓ1(W(G))=exp(M4+o(1)).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f=\sum _{W \in {\mathcal {W}}(G)}{z(W)1_W}\quad \text { and }\quad \Vert z\Vert _{\ell _1({\mathcal {W}}(G))} =\exp (M^{4+o(1)}). \end{aligned}$$\end{document}

and it follows from this and Parseval's theorem (see (6.1) in §6 if unfamiliar) that Our main result is the following weak converse. Since there are infinitely many primes it follows that for all n ∈ N there is some prime N ≥ 4n + 2 such that G := Z/N Z contains a set A of size 2n + 1 with 1 A A(G) ≤ 4 π 2 log n + O (1). Since N is prime we see that any representation of 1 A in terms of a function z of the required type must have z 1 (W(G)) ≥ |A| from which we get the result.
In fact Fejér's calculation in [11, (17.)] includes a determination of the O(1) term in the form c 0 + c 1 n + o n→∞ (n −1 ) so that the constant behind the can be computed rather accurately if desired, and Watson in [39] went even further with the asymptotic expansion using Szegő's beautiful formula for the Lebesgue constants in [34].
Proving our main result in the setting of general finite Abelian groups rather than Abelian groups of bounded exponent adds a number of difficulties. To help understand the overarching method we have presented Theorem 1.1 in the case when G is a group of exponent 2 in [30], where the simplifications also lead to a better bound. We state this result explicitly in §11 along with some results from other classes of group where more can be said.

Applications and Connections
Although some similarity may already be clear at this stage, we explicitly connect our work to Cohen's idempotent theorem in §12. One of the applications of this is to describe the algebra homomorphisms L 1 (G 1 ) → M(G 2 ) where G 1 and G 2 are locally compact Abelian groups. The rough idea is to note that such a map must arise as the pullback of a function between the dual groups whose graph has small algebra norm. The details may be found in [26, §4.1.3].
Wojciechowski [40], and then Czuron and Wojciechowski [9], made use of quantitative information from the idempotent theorem to strengthen consequences of the results above about non-existence of algebra homomorphisms into 'local' results about the norms of maps between finite dimensional subspaces. Stronger quantitative information in the present paper can be inserted directly to give stronger information there.
As a last connection to other work we mentioned that there is a quantitative connection between the coset ring (defined just before Theorem 12.1) and the stability ring of Terry and Wolf [36,37].

Outline of the Paper
Before moving on to the rest of the paper we should discuss the structure and notation, and a little about the contribution. The overarching structure is the same as that of [16]. In §2, §3, §4, §5 and §6, we set up the basic background theory we shall need which is Conversely we have Ruzsa's covering lemma.

Lemma 2.3 (Ruzsa's covering lemma) Suppose that A, B ⊂ G for some B = ∅. Then
Proof Suppose that X ⊂ A is maximal such that for every distinct x, x ∈ X we have (x + B) ∩ (x + B) = ∅. It then follows that if x ∈ A \ X , there is some x ∈ X such that ( In the light of Lemma 2.1 part (iv) above, for sets S, T ⊂ G with 0 G ∈ T it is natural to define the difference covering number of S by T to be where Ab denotes the category of Abelian groups and Hom(G, H ) is the set of homomorphisms between G and H . As before we often omit the subscript if the underlying group is clear. Again, since 0 G ∈ T the minimum above is well-defined, and if S is non-empty then C G (S; T ) ≥ 1.
For our purposes difference covering numbers turn out to behave slightly better than covering numbers.
The map φ × ψ is a group homomorphism G → H × H (defined by x → (φ(x), ψ(x))). Moreover, By the definition of the difference covering number and Lemma 2.1 (ii) we have that Part (ii) is proved.
Thirdly, let φ : G → G be the identity homomorphism, U := S and V := T so that S ⊂ φ −1 (U ) and φ −1 (V − V ) ⊂ T − T . It follows that and (iii) is proved.
Finally, let φ ∈ Hom(G, H ) and U , V ⊂ H be such that S ⊂ φ −1 (U ) and φ −1 (V − V ) ⊂ T and C H (U ; V ) = C G (S; T ). Then by Lemma 2.1 (i) and (iv) we see that This gives (iv).
It will also be useful to have a version of Ruzsa's covering lemma for difference covering numbers. Proof Let H be an Abelian group, φ ∈ Hom(G, H ) and U , V ⊂ H be such that φ −1 (U ) ⊃ X − X and φ −1 (V − V ) ⊂ B. By Ruzsa's covering lemma (Lemma 2. 3) we see that there is some set T with |T | ≤ m G (A + X ) m G (X ) and A ⊂ T + X − X .
Let U := φ(T )+U so that C H (U ; V ) ≤ |T |C H (U ; V ). On the other hand φ −1 (U ) ⊃ T + X − X ⊃ A and the result follows.

Bohr Systems
Bohr sets interact particularly well with covering numbers and difference covering numbers. We write · for the map S 1 → [0, 1 2 ] defined by It is easy to check that this is well-defined and that the map (z, w) → zw −1 is a translation-invariant metric on S 1 . Given a set of characters on G, and a function δ : → R >0 , then we write and call such a set a (generalised 2 ) Bohr set.
In fact we shall not so much be interested in Bohr sets as families of Bohr sets. A Bohr system is a vector B = (B η ) η∈(0,1] for which there is a set of characters and a function δ : → R >0 such that We say that B is generated by ( , δ) and, of course, the same Bohr system may be generated by different pairs. This definition is motivated by that of Bourgain systems [16, Definition 4.1], although it is in some sense 'smoother'. (In this paper what we mean by this is captured by Lemma 3.4 which does not hold for Bourgain systems. ) We first record some trivial properties of Bohr systems; their proof is left to the reader.

Lemma 3.1 (Properties of Bohr systems) Suppose that B is a Bohr system. Then
[16, Definition 4.1] took the approach of axiomatising these properties along with something called dimension. In that vein we define the doubling dimension of a Bohr system B to be It may be instructive to consider two examples.
and so for all x, y ∈ B 1 we have x − y ∈ B 1 and so there is some subgroup H ≤ G such that B 1 = H . We show by induction that for each i ∈ N 0 the set B 2 −i contains a translate of H , from which the result follows since 0 G ∈ B 2 −i .
Turning to the induction: the base case of i = 0 holds trivially. Suppose that B 2 −i contains a translate of H . Then there is some set contains a translate of H as required and the first result is proved.
In the other direction, simply let := {γ : γ (x) = 1 for all x ∈ H } and let δ be the constant function 1/|G|. Writing B for the Bohr system generated by and δ we see that H ⊂ B η for all η ∈ (0, 1]. On the other hand if x ∈ B 1 then |G| γ (x) < 1 and It follows that 2π |G| γ (x) ∈ 2π Z and hence |G| γ (x) ∈ Z. We conclude that γ (x) = 0 and hence γ (x) = 1 for all x ∈ B 1 and γ ∈ . It follows that B 1 = H and hence B is a constant vector by nesting. It remains to note that C(H ; H ) = 1 and so dim * B = log 2 1 = 0 as claimed.
We say that a Bohr system B has rank k if it can be generated by a pair ( , δ) with | | = k.

Lemma 3.3 (Rank 1 Bohr systems)
Suppose that B is a rank 1 Bohr system. Then dim * B ≤ log 2 3.

Lemma 3.4
Suppose that B is a Bohr system and w(B) < 1 4 . Then To prove this we shall use the following trivial observation.

Proof of
In the other direction, since w(B) < 1 4 there is a pair ( , δ) generating B such that δ ∞ ( ) < 1 4 . . Since ηk ≤ 1 and each x ∈ X has x ∈ B η we conclude (by sub-additivity) that kx ∈ B ηk , and hence there is some z(x) ∈ Z such that kx ∈ z(x) + B 1 4 ηk . Suppose that z(x) = z(y) for x, y ∈ X . By sub-additivity and nesting we have Suppose that γ ∈ . Then we have just seen that γ (x − y) < 2 k δ(γ ) < 1 2k (since δ(γ ) < 1 4 ) and so by the Observation we see that Dividing by k and noting that γ was an arbitrary element of it follows that x−y ∈ B 1 2 η and hence x = y. We conclude that the function z is injective and hence |X | ≤ |Z | ≤ C( ). Finally, if X is maximal with the given property then for any y ∈ B η either y ∈ X and so y ∈ X + B 1 2 η or else there is some x ∈ X such that y ∈ x + B 1 2 η . It follows that and the left hand inequality is proved given the upper bound on |X |.
We can make new Bohr systems from old by taking intersections: given Bohr systems B and B we define their intersection to be Writing B(G) for the set of Bohr systems on G we then have a lattice structure as captured by the following trivial lemma.

Lemma 3.5 (Lattice structure) The pair (B(G), ∧) is a meet-semilattice, meaning that is satisfies
Proof The only property with any content is the first, the truth of which is dependent on the slightly more general definition of Bohr set we made. Suppose that B is generated by ( , δ) and B is generated by ( , δ ). Then consider the Bohr system B generated by It is easy to check that B = B ∧ B and hence B ∧ B ∈ B(G). The remaining properties are inherited pointwise from the meet-semilattice (P(G), ∩), where P(G) is the power-set of G, that is the set of all subsets of G.
As usual this structure gives rise to a partial order on Another way we can produce new Bohr systems is via dilation: given a Bohr system B and a parameter λ ∈ (0, 1], we write λB for the λ-dilate of B, and define it to be the vector We then have the following trivial properties. The doubling dimension interacts fairly well with intersection and dilation and it can be shown that for Bohr systems B, B and λ ∈ (0, 1]. (The first of these is trivial; the second requires a little more work.) The big-O here is inconvenient in applications and to deal with this we define a variant which is equivalent, but which behaves a little better under intersection. The dimension of a Bohr system B is defined to be Proof First, from Lemma 2.4, part (ii) we have Hence by Lemma 2.1 part (iii) and the definition of doubling dimension we have and so dim B ≤ 2 dim * B as claimed.
As well as the various notion of dimension, Bohr systems also have a notion of size relative to some 'reference' set. Very roughly we think of the 'size' of a Bohr system B relative to some reference set A as being C (A; B 1 ). This quantity is then governed by the following lemma.
(ii) (Size and non-triviality) If C (A; B 1 ) < |A| then there is some x ∈ B 1 with x = 0 G .
Proof By symmetry and sub-additivity of Bohr sets we see that (λB) 1 Write r for the largest natural number such that 2 r λ ≤ 1. By Lemma 2.1 part (iii) we see that where the last inequality is by Lemma 2.4 part (iv) and the first inequality in Lemma 3.7 part (iii). The first part follows.
By Lemma 2.4 part (iv) we then see that C(A; B 1 ) ≤ C (A; B 1 ) < |A|. It follows that there is some set X with |X | < |A| such that A ⊂ X + B 1 whence |A| ≤ |X ||B 1 | < |A||B 1 | which implies that |B 1 | > 1 and hence contains a non-trivial element establishing the second part.

Measures, Convolution and Approximate Invariance
Given a finite set X we write C(X ) for the complex-valued functions on X . (We think of X as a discrete topological space and these functions as continuous with an eye to §12, hence the notation.) Further, given a probability measure μ on X and a set S with μ(S) > 0, we write μ S for the probability measure induced by Moreover, if S is a non-empty subset of G then we write m S for (m G ) S . (Note that this notation is consistent since m G = (m G ) G .) Below we shall define various notation for functions and for measures. Since G is finite we can associate to any measure μ on G a function y → μ({y}). The notational choices we make are designed to be compatible between these two different ways of thinking about measures hence the slightly unusual choice in (4.1).
Given f ∈ C(G) and an element x ∈ G we define We write M(G) for the space of complex-valued measures on G and to each μ ∈ M(G) associate the linear functional The functionals defined above are all linear functionals by the Riesz Representation Theorem [26, E4], though of course it is rather simple in our setting of finite G. Given μ ∈ M(G) we define τ x (μ) to be the measure induced by, andμ to be the measure induced by Given f ∈ L ∞ (G) and μ ∈ M(G) we define and for a further measure ν ∈ M(G) we define the convolution of μ and ν, denoted μ * ν, to be the measure induced by This operation makes M(G) into a commutative Banach algebra with unit; for details see [26, §1.3.1]. This notation all extends in the expected way to functions so that if f ∈ L 1 (m G ) thenf is defined point-wise by and given a further g ∈ L 1 (m G ) we define the convolution of f and g to be f * g which is determined point-wise by This can be written slightly differently using the inner product on L 2 (m G ).
Given a Bohr system B we say that a probability measure μ on G is Bapproximately invariant if for every η ∈ (0, 1] there are probability measures μ + η and μ − η such that It may be worth remembering at that for two measures ν and κ we say ν ≥ κ if and only if ν − κ is non-negative.
To motivate the name in this definition we have the following lemma where we recall that μ := d|μ|.

Lemma 4.1 Suppose that B is a Bohr system and μ is B-approximately invariant.
Then for all η ∈ (0, 1] we have It follows that The Jordan decomposition theorem tells us that there are two measurable sets P and N (which together form a partition of G) such that τ x (μ) − μ is a non-negative measure on P and a non-positive measure on N . We conclude that since μ + η and μ − η are probability measures and N P = G. The result is proved.
This can be slightly generalised in the following convenient way.

Lemma 4.2 Suppose that B is a Bohr system and μ is a B-approximately invariant probability measure. Then
Proof Simply note that by the triangle inequality and Lemma 4.1.
Approximately invariant probability measures are closed under convolution with probability measures.

Lemma 4.3
Suppose that B is a Bohr system, μ is a B-approximately invariant probability measure, and ν is a probability measure. Then μ * ν is a B-approximately invariant probability measure.
Since ν is a probability measure we can integrate the above inequalities to get

Proposition 4.4 Suppose that B is a Bohr system, and X is a non-empty set with
Then there is a λB-approximately invariant probability measure with support contained in X + B 1 for some 1 ≥ λ ≥ 1 24 log 2K .
Proof Let C := 24 and λ := 1/C log 2K . Note that K ≥ 1 and so λ < 1/4. Suppose that for all κ ∈ 1 4 , 3 4 there is some δ κ ∈ (0, λ] such that , and note that κ I κ ⊃ 1 4 , 3 4 . By the Vitali covering lemma 3 we conclude that there is a sequence κ 1 < · · · < κ m such that the intervals Since the intervals ( is an increasing sequence, and δ κ 1 , δ κ m ≤ λ < 1 4 we see that and hence This is a contradiction and so there is some κ ∈ 1 4 , 3 4 such that Let μ be the uniform probability measure on X + B κ , and for each η ∈ (0, 1] let μ − η be the uniform probability measure on X + B κ−λη and μ + η be the uniform probability measure on X + B κ+λη . If x ∈ (λB) η then x ∈ B λη and so Similarly The result is proved.
For applications it will often be useful to have the following corollary.
However, by sub-additivity of Bohr sets B 1 Thus given the definition of doubling dimension and the first inequality in Lemma 3.7 part (iii) we see that By Proposition 4.4 applied to X and B there is a λB -approximately invariant probability measure μ with support in The result follows since λ ≥ 1/24 log 2 2d+1 and λB = λ 2 B.

Approximate Annihilators
We shall understand the dual group of G through what we call 'approximate annihilators', though this nomenclature is non-standard. Given a set S ⊂ G and a parameter ρ > 0 we define the ρ-approximate annihilator of S to be the set Approximate annihilators enjoy many of the same properties as Bohr sets as we record in the following trivial lemma (an analogue of Lemma 3.1).

Lemma 5.1 (Properties of approximate annihilators) Suppose that S is a set. Then
Approximate annihilators and approximately invariant measures interact rather well as is captured by the following version of [14,Lemma 3.6]. To state it we require the Fourier transform extended to measures: for μ ∈ M(G) we define
In the more general topological setting where G is not assumed finite, approximate annihilators form a base for the topology of the dual group [ A number of elements of this paper would be neater if our Bohr sets were replaced by (a suitable generalisation of) sets of the form given in (5.1). The only benefit we know of arising from our choice is that the proof of Lemma 3.4 is slightly easier for vectors of Bohr sets. For us the duality in [26, Theorem 1.2.6] is captured in the following lemma.

Lemma 5.3 (Duality of Bohr sets and approximate annihilators)
(i) If X is a non-empty subset of G and ∈ (0, 1] then (ii) if is a non-empty set of characters of G and δ : → R >0 then On the other hand z ≤ 1 2 for all z ∈ S 1 and 2 − 2 cos 2π z = |z − 1|.
It follows that The result is proved once we disentangle the meaning of the two claims.
The following is [35,Proposition 4.39] extended to two sets. The proof is the same.

Lemma 5.4 Suppose that S, T are non-empty sets such that m G
and the result is proved.

Fourier Analysis
In this section we turn our attention to the Fourier transform itself. First we have the Fourier inversion formula [26, Theorem 1.
Since G is finite this is a purely algebraic statement which can be easily checked. It can be used to prove Parseval's theorem [26, Theorem 1.
One of the key uses of Bohr sets is as approximate invariant sets for functions.

Lemma 6.1 Suppose that is a set of k characters. Then there is a Bohr system B with
Proof For each γ ∈ let B (γ ) be the Bohr system with frequency set {γ } and width function the constant function 1 2 and put B := γ ∈ B (γ ) . (Equivalently, let B be the Bohr system with frequency set and width function the constant function 1 2 By Lemma 3.3 (and the second inequality in Lemma 3.7 part (iii)) we have dim B (γ ) = O(1) and by Lemma 3.7 part (i) we conclude that dim B = O(k).
Now, suppose that f is of the given form, meaning supp f ⊂ and f ∈ A(G). Then by Fourier inversion we have On the other hand the second part of Lemma 5.3 tells us that this supremum is at most when x ∈ B 1 π and the result is proved.
The next result is a variant of [7, Lemma 3.2] proved using their beautiful method.

Lemma 6.2 Suppose that B is a Bohr system, μ is B-approximately invariant, g ∈
A(G), and p ∈ [1, ∞) and ∈ (0, 1] are parameters. Then there is a Bohr system B ≤ B such that for any A ⊂ G we have and Proof We may certainly suppose that g ≡ 0 so that g A(G) > 0 (or else simply take B := B and we are trivially done). Consider independent identically distributed random variables X 1 , . . . , X l taking values in L ∞ (G) with Note that this is well-defined since 0 < g A(G) < ∞. Moreover, by the Fourier inversion formula, we have Regarding the variables X i (x) − g(x) g −1 A(G) as elements of L p (P l ) and noting, further, that we can apply the Marcinkiewicz-Zygmund inequality (see e.g. We integrate the above against μ + 1 (recall this is one of the family of measures provided by the hypothesis that μ is B-approximately invariant) and rearrange so that . Now, take l = O( −2 p) such that the right hand side rescaled is at most It follows that there are characters γ 1 , . . . , γ l such that Since f A(G) ≤ g A(G) (by the triangle inequality) we may apply Lemma 6.1 to the set of character {γ 1 , . . . , γ l } to get a Bohr system B with C (G; . If x ∈ B 1 then by the approximate invariance of μ we have τ x (μ) ≤ 2μ + 1 and μ ≤ 2μ + 1 , and so by the triangle inequality we have We conclude that and note by Lemma 3.7 parts (i) and (ii), and the earlier bound on dim B that and by Lemma 2.4 part (ii) and Lemma 3.8 part (i) and the bounds on B we have The result is proved.

Quantitative Continuity
It is well known that if G is a locally compact Abelian group and f ∈ A(G) then f is uniformly continuous. If G is finite then this statement has no content-every function on G is uniformly continuous-but in the paper [14], Konyagin and Green proved a statement which can be thought of as a quantitative version of this fact which still has content for finite Abelian groups. The main purpose of this section is to prove the following result of this type using essentially their method. and and a B -approximately invariant probability measure μ and a probability measure ν supported on B κ such that We shall prove Proposition 7.1 iteratively using the following lemma (which is, itself, proved iteratively).

Lemma 7.2
Suppose that B is a Bohr system of dimension at most d (for some d ≥ 1), ν is a B-approximately invariant probability measure, μ is a probability measure supported on a set X , f ∈ A(G) and δ, η ∈ (0, 1] and p ≥ 1 are parameters. Then at least one of the following is true: Proof Since the hypotheses and conclusions are invariant under translation by x it suffices to prove that if then we are in the second case of the lemma. Let κ := log 2 8δ −1 −1 for reasons which will become clear later; at this stage it suffices to note that κ ∈ (0, 1/2]. Define δ i := (1−κ) i δ for integers i with 0 ≤ i ≤ κ −1 and put g 0 := f − f * μ. Suppose that we have defined a function g i such that (G) and g i = g 0 * μ i for some probability measure μ i . By taking μ 0 to be the delta probability measure assigning mass 1 to 0 G , we see from (7.1) that g i satisfies these hypotheses for i = 0. By Lemma 6.2 applied to the function g i , the Bohr system B and measure ν with parameters p and i := κ g i L p (ν) g i

By Corollary 4.5 applied to B
1 . Integrating (and applying the integral triangle inequality) we conclude that and so by the triangle inequality and hypothesis on g i we have Put g i+1 := g i * ν (i) and μ i+1 = μ i * ν (i) . If g i+1 A(G) ≤ 2 1−(i+1) f A(G) then repeat; otherwise terminate the iteration. Since κ ≤ 1 2 and x → (1 − x) x −1 is monotonically decreasing for all x ∈ (0, 1] we see that if i ≤ κ −1 then Given our choice of κ we see that By choice of i, construction of μ i , and definition of g 0 we have (where we use the fact that f * μ(γ ) = f (γ ) μ(γ )) If γ ∈ G is such that |1 − γ (x)| < 2 −5 δ for all x ∈ X , then by the triangle inequality |1 − μ(γ )| ≤ 2 −5 δ, and hence the second sum on the left is at most 2 −5 δ f A(G) . Since |1 − μ(γ )| ≤ 2 by the triangle inequality, the third sum on the left is at most 2 f A(G) · 2 −6 δ, and so by the triangle inequality we have Put B := λ i−1 B (i−1) and apply Lemma 5.2 to ν (i−1) and B with parameters 2 −6 δ and η to see that Writing ρ := 2 −i−1 = (δ) and recalling that |1 − μ(γ )|| ν (i−1) (γ )| ≤ 2 by the triangle inequality we have It remains to note that i−1 > κδ i−1 2 i−2 = (κδρ −1 ) and so by Lemma 3.7 part (ii), and (7.3) we see that dim B satisfies the claimed bound. Finally, by Lemma 3.8 part (i), (7.2), (7.3), and the lower bound on λ i we have for any A ⊂ G from which the lemma follows.
Proof of Proposition 7.1 We proceed iteratively constructing Bohr systems (B (i) ) J i=0 and reals We  1 and f with parameters δ and 2 −5 δ (and p).
Suppose the conclusion of the second case of Lemma 7.2 holds. Then there is some ρ i = (δ) and a Bohr system and for any A ⊂ G we have However, for any A ⊂ G we have by Lemma 3.8 part (i). Thus for any A ⊂ G we have Additionally we have 1) and we get (iii). Moreover, by the order preserving nature of dilation and the fact that 2 −12 δ ≤ 1 and λ i κλ i ≤ 1; it follows that we have (ii). Now, Lemma 3.7 part (ii) and (7.5) gives from which we get (v). Finally, Lemma 3.8 part (i) tells us that for any A ⊂ G we have from which we get (iv).
In the light of (ii) we see that B It follows that after i steps we have and hence j≤i ρ j ≤ 1.
Since ρ j = (δ) we conclude that we must be in the first case of Lemma 7.2 at some step J = O(δ −1 ) of the iteration. In light of (v) we see that It then follows from (7.6) that for any A ⊂ G we have We now put B := λ J B (J ) , μ := μ J and ν := ν J , so that By Lemma 3.8 part (i) we see that for any A ⊂ G we have and by Lemma 3.7 part (ii) we have The result follows.

A Freiman-Type Theorem
The purpose of this section is to prove the following proposition, which is a routine if slightly fiddly variation on existing material in the literature.

Proposition 8.1 Suppose that A is non-empty and m G (A + A) ≤ K m G (A). Then there is a Bohr system B with
such that for any probability measure β supported on B 1 .
The proposition itself is closely related to Freiman's theorem and we refer the reader to [35,Chapt. 5] for a discussion of Freiman's theorem. For our purposes there are two key differences: (i) Freiman's theorem is usually only stated with the first two conclusions. It is possible to infer the fact that for any probability measure β supported on B 1 from the bound on C (A; B 1 ), and the fact that one can do better and get (8.1) in this sort of situation is an unpublished observation of Green and Tao. (ii) Freiman's theorem also produces a coset progression rather than a Bohr system. A set M is a d-dimensional coset progression if there are arithmetic progressions P 1 , . . . , P d and a subgroup H such that M = P 1 + · · · + P d + H . This definition was made by Green and Ruzsa in [15] when they gave the first proof of Freiman's theorem for Abelian groups. The conclusion of Freiman's theorem then is that there is a coset progression M with

and the challenge is to identify good estimates for the O K (1)-terms.
For us it is the quantitative aspects of Proposition 8.1 that are important. The quantitative aspects of Freiman's theorem are surveyed in [29], and primarily arise from the quantitative strength of the Croot-Sisask Lemma (in particular the m-dependence in [6, Proposition 3.3]), but also some combinatorial arguments of Konyagin [21] discussed just before [29,Corollary 8.4]. Conjecturally all the big-O terms should be O(log 2K ), though the proof below does not come close to that. It could probably be tightened up to same on the power of log(2 log 2K ) in the first two estimates above, at least reducing the 4 to a 3 but quite possible further.
We shall prove Proposition 8.1 as a combination of the next three results which we shall show in §8.1, §8.2, and §8.3 respectively. We say that a set X has relative polynomial growth of order d if m G (n X) ≤ n d m G (X ) for all n ≥ 1.
The first result can be read out of the proof of [29, Proposition 2.5] and essentially captures the power of the Croot-Sisask Lemma for our purposes.

Lemma 8.2 Suppose that A is non-empty with m G (A + A) ≤ K m G (A).
Then there is a symmetric set X containing the identity of relative polynomial growth of order O(log 3 2K (log(2 log 2K )) 3 ) and and some naturals m = (log 2K (log(2 log 2K ))) and r = O(log(2 log 2K )) such that m X ⊂ r (A − A).
The second result is one we have already touched on and captured a key insight of Green and Ruzsa in [15] that allows passage from relative polynomial growth to structure.

Lemma 8.3 Suppose that X is a symmetric non-empty set with relative polynomial growth of order d ≥ 1. Then there is a Bohr system B with dim B = O(d) and m G (B 1 ) = d O(d) m G (X ).
such that X − X ⊂ B 1 .
Finally the last lemma is a development of a result of Bogolioùboff [2] revived for this setting by Ruzsa [27], and then refined by Chang [5].

Lemma 8.4 Suppose that A is a non-empty set, B is a Bohr system and μ is a Bapproximately invariant probability measure, S ⊂ B 1 has μ(S) > 0, and L, nonempty, is such that
With these results in hand we can turn to proving the main result of the section.
Proof of Proposition 8. 1 We apply Lemma 8.2 to A to get a non-empty symmetric set X of relative polynomial growth of order O(log 2K log(2 log 2K )) 3 with m G (X ) ≥ exp(−O(log 2K log(2 log 2K )) 3 )m G (A), (8.2) and natural numbers m = (log 2K log(2 log 2K )) and r = O(log(2 log 2K )) such that m X ⊂ r (A − A). By Lemma 8.3 there is a Bohr system B with X − X ⊂ B 1 such that By nesting of Bohr we have that By Corollary 4.5 there is a probability measure μ and a Bohr system B = λB for some λ = ((1 + dim B ) −1 ) such that μ is supported on B 1 and μ is Bapproximately invariant. By Lemma 3.8 part (i) (with reference set X − X ) we have By the second inequality in Lemma 3.7 part (iii) and the definition of dimension there is a set T with It follows from nesting of Bohr sets that Inserting the upper bound for m G (B 1 ) and the upper bound for |T |, it follows that there is some x such that μ(x + X ) ≥ exp(−O(log 3 2K (log(2 log 2K )) 4 )).
Now, put S := x + X and note from Plünnecke's inequality that Given the lower bound on m and upper bound on r it follows that there is some Putting L := A − A + l S it follows by the Cauchy-Schwarz inequality that By Lemma 8.4 (with reference set X − X ) we then see that there is a Bohr system such that Since 0 G ∈ X we see that X ⊂ r (A − A) and hence by Lemma 2.5 and Plünnecke's inequality (and (8.2) and (8.3)) we have Finally, if β is supported on B 1 then from which the final bound follows by Plünnecke's inequality.

Croot-Sisask Lemma Arguments
The aim of this section is to prove the following lemma.

Lemma (Lemma 8.2) Suppose that A is non-empty with m G (A + A) ≤ K m G (A).
Then there is a symmetric set X containing the identity of relative polynomial growth of order O(log 3 2K (log(2 log 2K )) 3 ) and and some naturals m = (log 2K (log(2 log 2K ))) and r = O(log(2 log 2K )) such that m X ⊂ r (A − A).
The material follows the proof of [29, Proposition 8.5] very closely, though we shall need some minor modifications. We start by recording two results used to prove that proposition.

Corollary 8.5 ([29, Corollary 5.3]) Suppose that X ⊂ G is a symmetric set and m G ((3k + 1)X ) < 2 k m G (X ) for some k ∈ N. Then X has relative polynomial growth of order O(k).
This is just a variant of Chang's covering lemma from [5] (see also [35,Lemma 5.31]).
This captures the content of the Croot-Sisask Lemma [6, Proposition 3.3] for our purposes.
We shall also need a slight variant of [29,Proposition 8.3]. Proof Let f := 1 A+S and apply the Croot-Sisask lemma (Lemma 8.6) with parameters η and p (to be optimised later) to get a symmetric set X containing the identity It follows by the triangle inequality that Taking an inner product with m A we see that for all x ∈ X we have where p is the conjugate exponent to p. Now Thus We take p = 2 + log K , and then η = (m −1 ) such that the term on the right is at most 1/2 to get the desired conclusion.
The above proposition is almost all we need for our main argument and it can be used in the proof of Lemma 8.2 below to give a result with only slightly weaker bounds. However, we shall want a slight strengthening proved using the aforementioned idea of Konyagin [21].

Proof Define sequences
by Plünnecke's inequality we have K i ≤ K 2r i . We proceed inductively to define sequences of non-empty sets (S i ) i≥0 and (T i ) i≥0 with We shall establish the following properties inductively for all i ≥ 0.
(i) S i and T i are symmetric sets containing the identity such that (ii) and (iii) and (iv) and We initialise with S 0 := A − A and T 0 := A − A so that S 0 and T 0 are symmetric sets containing the identity (since A is non-empty) and whence (i) holds. Moreover, by Plünnecke's inequality we have so that (ii) holds. Suppose that we are at stage i of the iteration. Apply Proposition 8.7 to the sets A, S i , and T i with parameter m i . This produces a symmetric set T i+1 containing the identity such that First note that given the definition of m i , r i and r i+1 we have and so we get (iv). The second part of (8.4) ensures (iii). Moreover, we have By the pigeon-hole principle there is some non-negative integer l i ≤ m i /s − 1 such that which is a symmetric set containing the identity since both T i+1 and A − A are. Since 0 G ∈ T i+1 and l i ≤ m i /s − 1 we have which gives (i). Moreover, from (8.5) we have so that (ii) holds. Let i ≥ 1 be maximal such that 2r i−1 + 1 ≤ r (possible since r ≥ 3 = 2r 0 + 1, so that  We get a natural m = (rs log 2K ) and a symmetric set S containing the identity such that

m X ⊂ r (A − A) and m G (X ) ≥ exp(−O(s 2 r 3 log 3 2K ))m G (A).
Let k := m 3 . By Plünnecke's inequality we have For s = O(1) sufficiently large the right hand side is strictly less than 2 k (since X is non-empty) and hence we can apply Corollary 8.5 to see that X has relative polynomial growth of order O((log(2 log 2K )) 3 log 3 2K ). The result is proved.

From Relative Polynomial Growth to Bohr Sets of Bounded Dimension
The next proposition is routine with the core of the argument coming from [15].

Lemma (Lemma 8.3) Suppose that X is a symmetric non-empty set with relative polynomial growth of order d ≥ 1. Then there is a Bohr system B with
such that X − X ⊂ B 1 .
Proof Let m = O(d log 2d) be a natural number such that m d m−1 ≤ 3 2 . Since X has relative polynomial growth of order d we see by the pigeonhole principle that there is some 2 ≤ l ≤ m such that Let := 1/2 18 d 2 (the reason for which choice will become clear later) and write so that by Lemma 5.4 (applicable since l ≥ 2) we have that Let δ : → R >0 be the constant function taking the value 2 −4 and B be the Bohr system with frequency set and width function δ. By the first part of Lemma 5.3 we and hence | β(γ )| ≥ 1 2 . We conclude that But, by Parseval's theorem and Hölder's inequality we have that and so Now, note by sub-additivity and symmetry of Bohr sets and Ruzsa's Covering Lemma (Lemma 2.3) that for i ≥ 1 we have .
where the last inequality is from (8.6).
By averaging there is some 0 ≤ j ≤ J such that where the last inequality is from (8.8).
The result is proved.

Bogolioùboff-Chang
In the paper [2] Bogolioùboff showed how to find Bohr sets inside four-fold sumsets. The importance of this was emphasised by Ruzsa in [27] and refined by Chang in [5]. We shall need the following result in our work.

Lemma (Lemma 8.4) Suppose that A is a non-empty set, B is a Bohr system and μ is a B-approximately invariant probability measure, S ⊂ B 1 has μ(S) > 0, and L, non-empty, is such that
Proof Since μ is B-approximately invariant and μ is a probability measure, Lemma 4.3 tells us that μ * μ is B-approximately invariant. By Parseval's theorem we have Apply Lemma 6.2 to B, μ * μ, and 1 L * 1 −L with parameters p ≥ 2 and η ∈ (0, 1] to be optimised later. This gives us a Bohr system B with Since μ is non-negative we have and so there is a function f with 0 ≤ f ≤ μ(S) −2 point-wise such that ( f is the Radon-Nikodym derivative of μ S * μ S with respect to μ * μ.) Write p for the conjugate index of p (so 1 If we take p = 2 + 2 log μ(S) −1 then we see from Hölder's inequality that for all x ∈ B 1 we have By hypothesis it follows that for η = 1 2e we have However, the left hand side is 0 if The result is proved.

Arithmetic Connectivity
The basic approach of our main argument (captured in Lemma 10.2) is iterative and to make this work we need to consider not just integer-valued functions, but almost integer-valued functions. For ∈ (0, 1/2) we say that f : Since < 1/2 this actually means that f Z is uniquely defined.
When a function f has small algebra norm and is close to integer-valued, it turns out that f Z has a lot of additive structure. This is captured by a concept called arithmetic connectivity identified by Green in [16,Definition 6.4]. We shall need a slight refinement of this: for m, l ≥ 2 we say that a set A ⊂ G is (m, l)-arithmetically connected if for every x ∈ A m there is some σ ∈ Z m with σ m 1 ≤ l and |σ i | = 1 for at least two is such that The definition is perhaps a little odd. To help we present some simple examples we leave as exercises.
(i) A is (m, 1)-arithmetically connected for some m if and only if A = ∅. (Of course this is not a significant example and can easily be removed by simply restricting to m, l ≥ 2.) (ii) If every element of A has order 2 then A is (m, m + k)-arithmetically connected for some k ≥ 0 if and only if it is (m, m)-arithmetically connected. (iii) If A is a subgroup then x + y ∈ A for all x, y ∈ A and so A is (2, 2)-arithmetically connected. On the other hand, if G = Z and A = N then A is also (2, 2)arithmetically connected (for the same reason) but not 'close' to any subgroup. (iv) If A is a union of k cosets (of possibly different subgroups) then by the pigeonhole principle for any vector x ∈ A 2k+1 there are indices i < j < k such that x i , x j , x k are all in the same coset. It follows that x i + x j − x k is in that same coset and hence in A. We conclude that A is (2k + 1, 3)-arithmetically connected.
Arithmetic connectivity is related to additive structure by the following easy adaptation of [16, Proposition 6.5].

Lemma 9.1 Suppose that A is (m, l)-arithmetically connected (for m, l ≥ 2). Then
Proof First we count the number of σ ∈ Z m such that σ m 1 ≤ l. The number of ways of writing a total of r as a sum of m non-negative integers is r +m m . For each such σ we can choose the signs of the various integers in at most 2 l ways (since at most l of them are non-zero) and so the total number of σ ∈ Z m with σ m 1 ≤ l is at most It follows that there is such a σ ∈ Z m such that for at least m −O(l) |A| m vectors x ∈ A m we have σ · x ∈ A. Rewriting this we have Since The result now follows from Cauchy-Schwarz and Parseval's theorem which gives On the other hand additive connectivity is related to small algebra norm via the following result.

Proposition 9.2
There is an absolute constant C Mél > 0 such that the following holds. Suppose that g ∈ A(G) is -almost integer-valued for some ∈ (0, 1/2) and has g A(G) ≤ M for some M ≥ 1.
The proof of this owes a lot to [24, Lemme 1] of Méla, and we are grateful to Ben Green for directing us to that paper. Indeed, as noted in [16, §9] an example in Méla's paper shows that one cannot hope to weaken the requirement that ≤ exp(−C M)) to anything with C below a certain absolute threshold. One can also make use of the auxiliary measures [24,Lemme 4] constructed in Méla's paper to show that supp g Z is (O(M 2 log 2M), O(M log 2M))-arithmetically connected but for us this extra logarithm in the second parameter is worse than the benefit of a power saving in the first when we apply Lemma 9.1.
We write T n (x) for the Chebychev polynomial of degree n. Recall (from, for example, [43, §6.10.6]) that we have a formula for T n : the last form tells us immediately that T n L ∞ ([−1,1]) ≤ 1.
We shall be particularly interested in the Chebyshev polynomials of odd degree. Indeed, note from the above formula that if n = 2l + 1 for some non-negative integer l, then only the coefficients of odd powers of x are non-zero and In view of this we have Added to this information we shall need the following lemma.

Lemma 9.3
Suppose that m ∈ N, and l ∈ N 0 are parameters, g : G → C has support A and x ∈ G m is such that if σ ∈ Z m has σ m 1 ≤ 2l + 1 and σ · x ∈ A then |σ i | = 1 for at most one value of i. Then for every ω ∈ m ∞ with ω m ∞ ≤ 1 and 0 ≤ r ≤ l we have Proof We write C for the conjugation operator and note that by Fourier inversion we have Applying the triangle inequality we see that By the triangle inequality we have Moreover, and so 1 A (σ (π, ι) · x) = 0 unless |σ j (π, ι)| = 1 for at most one j ∈ [m]. It remains to bound from above the number of functions π : [2r +1] → [m] and ι : [2r +1] → {0, 1} such that |σ j (π, ι)| = 1 for at most one j ∈ [m]. Since |σ j (π, ι)| = 1 for at most one j it follows that the image of π has size at most r + 1, and hence the number of pairs (π, ι) is at mosst m r + 1 · (r + 1) 2r +1 · 2 2r +1 = exp(O(r + 1))(r + 1) r m r +1 .
Inserting this into (9.2) gives the result.
Proof of Proposition 9.2 Let A := supp g Z , and take l and m to be parameters to be chosen later. Suppose that A is not (m, 2l + 1)-arithmetically connected, so that there is some x ∈ A m such that for all σ ∈ Z m with σ m 1 ≤ 2l + 1 and |σ i | = 1 for at least two i ∈ [m], we have g Z (σ · x) = 0.
Our first task is to define ω ∈ m ∞ . With ω appropriately defined we shall put The function g Z is real and since x j ∈ A we see that |g Z (x j )| ≥ 1 for all j ∈ [m]. It follows that (i) either at least 1/3 of the indices j ∈ [m] have g Z (−x j ) = 0, in which case we set ω j = sgn g Z (x j ) for all these indices and ω j = 0 for all others, and get m j=1 (ii) or at least 1/3 of the indices j ∈ [m] have sgn g Z (x j ) = sgn g Z (−x j ), in which case we set ω j = sgn g Z (x j ) for all these indices and ω j = 0 for all others and get m j=1 (iii) or at least 1/3 of the indices j ∈ [m] have sgn g Z (x j ) = − sgn g Z (−x j ), in which case we set ω j = i for all these indices and ω j = 0 for all others and get m j=1 By construction ω m ∞ ≤ 1 and By Lemma 9.3 for every 1 ≤ r ≤ l we have subgroups H 1 , . . . , H l ≤ G, and functions z 1 : G/H 1 → Z, . . . , z l : G/H l → Z such that To do this we combine all our previous work into our key iterative lemma.

Lemma 10.2 Suppose that f ∈ A(G) is -almost integer-valued, f
Proof Apply Proposition 9.2 to f to get that the set A := supp f Z is (O(M 3 ), O(M))arithmetically connected (provided is sufficiently small). By Lemma 9.1 we see that It follows from the Balog-Szemerédi-Gowers Theorem that there is a set A ⊂ A such that to get a Bohr system B ≤ B with and a B -approximately invariant probability measure μ and a probability measure ν supported on B κ such that By the integral triangle inequality it follows that Since μ is B -approximately invariant and κ ≤ 1/2 it follows from Lemma 4.2 that for all y ∈ supp ν * ν we have By the triangle inequality we then have given the choices of δ and κ, and the upper bound on . We put k := ( f * μ) Z which will turn out to be the g Z in the conclusion. We establish the various properties in order.
Proof Since κ ≤ 1/2 and B ≤ B we see that supp ν * ν ⊂ B 1 , and hence by (10.1) that which contradicts the choice of p. It follows that k(x) = 0.
Proof Note that and so Write H for the group generated by B κ so that Lemma 2.2, Lemma 2.4 part (iv), and Lemma 3.8 part (i) tell us From the claims, k is H -invariant and so there is a well-defined function z : G/H → Z such that z(W ) = k(w) for all w ∈ W . Now we have from the claims that It remains to put g := f * ν * ν and note that g Z = k has the required properties. Moreover, since k is not identically 0 we see that and so f i+1 is i+1 -almost integer-valued.
Since f i A(G) ≥ 0 we must have ( f l ) Z ≡ 0 for some l ≤ M (1 + δ). But then provided is sufficiently small. The result follows since f Z is uniquely defined in this case and ( f i ) Z ≡ 0 when the iteration terminates.

Specific Classes of Groups
In this section we discuss work for specific classes of groups.

Groups of Bounded Exponent
In [13] Green set out a model setting for additive combinatorics. (See [41] for a recent perspective.) In this setting a number of arguments simplify and Theorem 10.1 could be proved for groups of bounded exponent without the need for any discussion of Bohr systems.
As mentioned in the introduction [30] carries out this simplification for finite groups of exponent 2-i.e. groups isomorphic to F n 2 for some n-though more general (Abelian) groups of bounded exponent are no harder. In certain regimes there are already stronger results, at least for indicator functions of sets. Indeed Shpilka, Tal, and Lee Volk established the following in [32]. While our aim is to avoid any sort of |G| dependence, it is worth noting that in the above theorem it is really rather mild.
It is also interesting that for this class of groups arithmetic progressions are no longer a limiting example-we do not have Proposition 1.2-and it might be that the bound on z 1 (W(G)) can be polynomial in M. Some efforts in this direction for particular classes of function can be found in work of Tsang, Wong, Xie and Zhang, in particular [38, Corollary 7].

Cyclic Groups of Prime Order
For cyclic groups of prime order there are a range of results by Konyagin and various authors. In particular the following is an easy consequence of [14, Theorem 1.3].

Theorem 11.3 Suppose that G = Z/ pZ and A ⊂ G has m G
The above bound becomes weaker quite quickly as A gets smaller, and Konyagin and Shkredov [22,23] have the following results to deal with this.
The arguments behind these results are not restricted to indicator functions of sets and the results themselves have been extended by Gabdullin in [12]; that paper also develops some higher dimensional analogues.
In Z/ pZ there are no non-trivial subgroups and so these three results can be combined to give the following. Note that this is already a strengthening of the main result of [16] in the particular case of groups of prime order, and this has been further strengthened by Schoen in [31] who showed the above with a bound of the form exp(M 16+o(1) ) by combining Konyagin and Shkredov's work more effectively.
In fact Konyagin and Shkredov's results are much sharper if one takes A to be sparse. For example, they combine to give the following. Theorem 11.7 (Konyagin-Shkredov) Suppose that G = Z/ pZ and A ⊂ G has 1 A A(G) ≤ M for some M ≥ 1 and |A| ≤ p 9/10 . Then there is some z : W(G) → Z such that (1) ). This is stronger than our main theorem in this particular case of small sets in groups of prime order.

Torsion-Free Groups
For a non-vacuous discussion of torsion-free groups we need to have a definition of A(G) for infinite groups. This is virtually the same, but see the start of §12 for the formal details. Konyagin [20] and McGehee, Pigno and Smith [25] resolved the Littlewood conjecture by proving the following in our language. In fact some work has been done on the constant behind the big-O term. Stegeman [33] and Yabuta [42] independently give a bound of the shape for some c < 1. It must be that c ≥ π −1 in view of the size of the Lebesgue constants (see [11, (16.)]).

Cohen's Idempotent Theorem
In this section we extend our work to locally compact Abelian groups; suppose that G is such. Then we write G for the (locally compact Abelian group [26, §1.2.6, Theorem A short calculation [26, §3.1.2] shows that if W ∈ W(G) then W ∈ A(G) and 1 W B(G) = 1. It follows from the triangle inequality for · B(G) that if A ∈ A(G) then ¬A ∈ A(G) since 1 ¬A = 1 G − 1 A ; and it follows from the sub-multiplicativity The coset ring of G is the intersection of all rings of sets on G containing W(G). This is a ring, and by the above is contained in A(G). Cohen's idempotent theorem is the following converse. To give a quantitative version of this we need a more constructive view of L(G). With an eye to our later results we take a slightly more complicated definition than one might at first choose.
Given H ≤ G and S ⊂ G/H we write S * := S ∪ ¬ S , that is the partition of G into cells from S and an additional cell that is everything else. We say that A has a (k, s)-representation if there are open subgroups H 1 , . . . , H k ≤ G, and sets S 1 ⊂ G/H 1 , . . . , S k ⊂ G/H k of size at most s such that A is the (disjoint) union of some cells in the partition 4 S * 1 ∧ · · · ∧ S * k . We write W k,s (G) for the set of sets with (k, s)-representations. It can be shown fairly directly that k W k,s (G) = L(G) for any s ∈ N, but as this also follows from what we are about to show we omit the details.
The triangle inequality and sub-multiplicativity of · B(G) gives that each cell in the partition has algebra norm at most (s + 1) k and there are at most (s + 1) k cells so 1 A B(G) ≤ (s + 1) 2k for all A ∈ W s,k (G). (12.1) We shall prove the following converse. 4 Recall that if P and Q are partitions of the same set then P ∧ Q := {P ∩ Q : P ∈ P, Q ∈ Q}. We shall prove this after the proof of the next result. The earlier work of this paper concerned integer-valued functions, not just {0, 1}valued functions, and we now turn to these. We say that f : We begin with a qualitative variant of our result, [1,Theorem]. This gives open subgroups S 1 , . . . , S k ≤ G; mutually orthogonal measures μ 1 , . . . , μ k ∈ M( G); natural numbers R i ; signs and ( i, j ) R i j=1 , and elements (x i, j ) R i j=1 such that If one wished to avoid appealing to Cohen's theorem in the proof above the key obstacle comes in §9. The concept of arithmetic connectivity extends easily enough to locally compact Abelian groups (using, e.g., the definition of B(G) developed by Eymard [10, (2.14) Lemme] for non-Abelian groups), but this does not lead to a statement about large energy directly because we do not yet have a natural measure with respect to which the support of f is positive but finite. Let S i := {W ∈ G/H i : z i (W ) = 0} for 1 ≤ i ≤ k and note that 1 A is constant on cells of the partition S * 1 ∧ · · · ∧ S * k , which gives the required result.

Corollary 12.4 Suppose that G is a locally compact Abelian group and f ∈ B(G) is integer-valued with f B(G) ≤ M. Then f has an (M, O M (1))-representation.
This is best possible in the first parameter of the representation as can be seen by considering a disjoint union of cosets of subgroups H 1 , . . . , H l ≤ G where |H i + H j : It is important to note that the error term is not monotonic in the M parameter and this is necessarily the case: consider A := G \ {0 G } for G a group whose order is a large prime. Then 1 A B(G) < 2 and so if we are to write A as a sum of indicator function of cosets of at most 1 A B(G) subgroups, then there can only be one subgroup and we can require arbitrarily many cosets of this as the prime p increases.
Apart from Cohen's original proof [8] of Theorem 12.1, which is the proof on which Rudin's [26,Chapter 3] is based, there are proofs of the idempotent theorem due to Amemiya and Itô [1] (shortening Cohen's original argument), and Host [18] also shortening Cohen's argument, but the main purpose of which is to beautifully extend it to non-Abelian groups.
As stated these results are trivial for finite groups and the arguments do not seem to immediately extend to give quantitative information. Both Amemiya and Itô's and Host's are very soft; Cohen's less so. That being said they do have non-trivial quantitative content in one respect and in particular they can all be used to prove the following theorem. This is slightly weaker than Corollary 12.4 since there are multiple functions with the same algebra norm. It is worth noting that one cannot guarantee equality in the right sum in (12.5) for finite groups unless l = 1-the example following Corollary 12.4 applies here too. This means that we have to relax the requirement that the underlying measures-that is the measures μ i such that f i (x) = γ (x)dμ i (γ ) are mutually orthogonal to simply a requirement that they are 'quite' orthogonal. In some respects this is what happens in our quantitative continuity argument in §7.