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An inverse theorem: when the measure of the sumset is the sum of the measures in a locally compact abelian group

We classify the pairs of subsets (A,B) of a locally compact abelian group satisfying m(A+B)=m(A)+m(B), where m is Haar measure. This generalizes a result of M. Kneser classifying such pairs under the additional assumption that G is compact and connected. Our proof combines Kneser's proof with arguments of D. Grynkiewicz, who classified the pairs of subsets (A,B) of abelian groups satisfying |A+B|=|A|+|B|, where |A| is the cardinality of A.

1. Introduction 1.1. Inverse theorems. Inverse theorems in additive combinatorics deduce properties of subsets A, B of an abelian group from some hypothesis on the sumset A + B := {a + b : a ∈ A, b ∈ B}. The hypothesis on A + B is often a bound on some quantity (perhaps cardinality, Haar measure, or some kind of density), in terms of the same quantities related to A and B. The books [11,12] give a comprehensive introduction to a large segment of this theory. A very simple example of an inverse theorem says that if A and B are finite, and |A + B| ≤ max{|A|, |B|}, where |S| is the cardinality of the set S, then one of A or B is a union of cosets of a subgroup H G, and the other is contained in a coset of H. Kneser [8] described the pairs (A, B) of Haar measurable subsets of a locally compact abelian group G with Haar measure m satisfying m * (A + B) < m(A) + m(B). Here m * denotes the inner measure corresponding to m, that is, m * (S) = sup{m(E) : E ⊂ S, E is compact}. If S ⊂ G and t ∈ G, then S + t := {s + t : s ∈ S}.  1.2. Summary of results. In this article, we remove the hypothesis that G is connected from Theorem 1.2. The weaker hypothesis permits new types of sur-critical pairs, as shown in §1. 6. Our main result is Theorem 1.4, which describes all surcritical pairs for an arbitrary compact abelian group. Weakening the compactness hypothesis on G to local compactness produces no new examples, as shown in §4.
While the conclusion of Theorem 1.4 is trivial under the hypothesis that G is finite, the article [2] gives a very detailed description of sur-critical pairs for finite groups. The following discussion summarizes the development of some inverse theorems for finite groups and how these relate to Theorem 1.4.
Context. In contrast to inverse theorems, direct theorems deduce properties of A + B from hypotheses on A and B. One of the earliest such results is the Cauchy-Davenport inequality, which states that |A + B| ≥ min{|A| + |B| − 1, p} whenever A, B ⊂ Z/pZ for some prime p; here |S| is the cardinality of the set S. The corresponding inverse theorem, due to Vosper, classifies the pairs (A, B) of subsets of Z/pZ satisfying |A+B| = |A|+|B|−1, when |A|+|B| < p: the equation holds if and only if A and B are arithmetic progressions with the the same common difference, or one of |A| = 1, |B| = 1 [14,13].
Generalizing Vosper's theorem (and strengthening the special case of Theorem 1.1 where G is discrete), Kemperman [7] described those pairs of finite subsets A, B of an abelian group which satisfy |A + B| < |A| + |B|. Recently, extensions of Vosper's and Kemperman's theorems for |A + B| < |A| + |B|, describing the case |A + B| = |A| + |B|, have appeared [3,4]. More recently, Grynkiewicz [2] gave a very detailed description of those pairs (A, B) of finite subsets of an abelian group G satisfying |A + B| = |A| + |B|. The description given in [2] is somewhat intricate, so we do not reproduce it here. In Theorem 1.4, we describe those pairs (A, B) of subsets of a locally compact abelian group G satisfying m(A + B) = m(A) + m(B). Concatenating Theorem 1.4 with the results of [7] and [2] yields a very precise description of pairs (A, B) of subsets of a locally compact abelian group satisfying m * (A + B) = m(A) + m(B), but we will not state this description explicitly. Theorem 1.4 was developed partly to answer Question 4.1 of [6].
Background. We assume knowledge of the theory of locally compact abelian groups, in particular the fact that Haar measure can be disintegrated over the cosets of a compact subgroup; a compact abelian group G has either a neighborhood base at the identity consisting of subgroups, or a surjective character χ : G → R/Z; and subgroups of G having positive Haar measure are open. The references [9], [5], and [1] each provide sufficient background, as does [8].
1.3. Terminology and notation. Throughout, G will denote a locally compact abelian group and m G will be its Haar measure. When there is no chance of confusion, we write m for m G . The term measurable in reference to a subset of G will always mean "lies in the completion of the Borel σ-algebra with respect to m." Haar measure will always be normalized for compact groups, so m(G) = 1 for such G. The symbol T will denote the group R/Z, and "an interval in T" means a set of the form [x, y] + Z, where x ≤ y < x + 1.
If S ⊂ G, S c will denote the complement G \ S. then (A, B) is called a sur-critical pair. We call a pair (A, B) satisfying m * (A + B) < m(A) + m(B) a critical pair. Lemma 4).
If H G is a compact open subgroup, φ H : G → G/H will denote the quotient map. We may identify subsets of G of the form A + H with subsets of G/H.
Much of our terminology is taken from [2]; when G is discrete, some of our definitions coincide with those from [2].
Periodicity. If A ∼ A + H for some some compact open subgroup H G, we call A periodic with period H. Otherwise, we call A aperiodic. Note that the assertion H(A) = {0} implies A is aperiodic, but an aperiodic A may have m H(A) > 0.
Extendibility and reducibility. Let A and B be measurable subsets of a compact abelian group G. We say that A is extendible with respect to B if there is a measurable set A ⊃ A with m(A ) > m(A) and m * (A + B) = m * (A + B). We say that the pair (A, B) is extendible if A is extendible with respect to B or B is extendible with respect to A. Otherwise, we say that (A, B) is nonextendible.
The group H is called a quasi-period of A. Note that A may have more than one quasi-period.
We say that a pair (A, B) of subsets of G has a quasi-periodic decomposition with respect to H if one (or both) of A or B is quasi-periodic with respect to H and the other is either contained in a coset of H or is quasi-periodic with respect to H.   Letting ε → 0, we get (1.1). The same argument applied to B and A + B implies (1.2) and (1.3). Theorem 1.4 will show that most sur-critical pairs are essentially regular, and will give a simple description of those sur-critical pairs which are not essentially regular.
1.5. Description of sur-critical pairs. Our main result, Theorem 1.4, describes the sur-critical pairs for a compact abelian group. Corollary 4.4 will show how sur-critical pairs for a locally compact abelian group G may be obtained from surcritical pairs in a compact quotient of G.   The labels (P), (E), (K), and (QP) stand for "periodic", "extendible," "Kneser," and "quasi-periodic," respectively.
This classification is a fairly straightforward application of Theorem 1.4, and we omit the proof.
Remark 2. Sur-critical pairs satisfying (QP) can be further described in terms of critical pairs. We first describe a process for producing sur-critical pairs satisfying (QP) from critical pairs, then we show that every sur-critical pair satisfying (QP) arises from this process.
Let G be a compact abelian group with finite quotient F = G/H. Let (A , B ) be a critical pair for F such that |A + B | = |A | + |B | − 1, and let a ∈ A , b ∈ B such that a + b is a unique expression element of A + B . Let    Examples arising from finite groups. If F is a finite group, every sur-critical pair for F satisfies (P), and every nonextendible sur-critical pair satisfies (P) but not (E) or (K). Every periodic sur-critical pair (A, B) for a group G has the form (φ −1 (C), φ −1 (D)), where (C, D) is a sur-critical pair for a finite group F, and φ : G → F is a homomorphism, so the periodic sur-critical pairs for an arbitrary group G are described in [2].
Complementary pairs and extendible pairs. Examples satisfying (E) may be constructed in a straightforward way from critical pairs in infinite compact abelian groups. Specifically, let G be an infinite compact abelian group with a proper compact open subgroup H. Let C, D ⊂ G/H satisfy |C + D| = |C| + |D| − 1, and let c 0 + d 0 be a unique expression element of C + D.
, and further insist that A 0 has nonempty intersection with every coset of every compact open subgroup K ≤ H.
Generically, a pair (A, B) so constructed will not satisfy (P) or (K), but will satisfy (E.2).
To form a specific example of the above construction, A quasi-periodic pair. This example will satisfy (QP) but not (P), (E), or (K).
Let G = Z 7 , the 7-adic integers with the usual topology. Define the set C ⊂ Z by . The description of A+B shows that H(A+B) = {0}, so (P) and (E) fail. Since G is totally disconnected, (K) cannot hold.
An irregular pair. Let G 0 be an infinite compact abelian group with Haar measure m 0 , and , but depending on our choice of A 0 and B 0 , A + B may not be measurable. If A 0 is a singleton, then B 0 may be an arbitrary measurable subset of G 0 .

Lemmas
We fix a compact abelian group G with Haar measure m. Unless stated otherwise, sets A and B are assumed to be subsets of G.
The second assertion follows by reversing the roles of A and B.
We will need a corollary of Theorem 1.1. We write C ⊂ m D or D ⊃ m C when m(C \ D) = 0.
Proof. For the first equality, note that z Proof of Claim 1. By (2.1) and the definition of B 0 we can write the measure of Collecting terms, we find Since each bracketed summand above is nonnegative, we have We now proceed based on whether m(B 0 ) < m(H).
Proof of Claim 2. If a, a ∈ A are such that a+B 0 ⊂ A +B and a +B 0 ⊂ A +B, then a+B 0 ∼ a +B 0 , by (2.1). Since B 0 is contained in a coset of H, this similarity implies a − a ∈ H. We conclude that A 0 is contained in a coset of H.

Proof. By Lemma 2.4 (A, B) has a quasi-periodic decomposition
We need only show that A 0 + B 0 is a unique expression element of A+B+H in G/H. Assuming otherwise, we find that A+B is periodic, contradicting our assumption. Proof. Let H be the quasi-period of (A , B ). We may assume A ⊂ A and B ⊂ B, since we can replace A and B with A ∩ A and B ∩ B while maintaining the hypotheses of the lemma -this follows from the irreducibility of (A, B) and (A , B ). Assume, without loss of generality, that A 1 = ∅. Define A 0 := A∩(A 0 +H) and Let a ∈ A, and observe that a A similar argument shows that B has a quasi-periodic decomposition with respect to H. The following lemma is standard; we include a proof for completeness. Lemma 2.9. If S ⊂ G is measurable and let Then m(E) = 0.

Proof. Write H for H(S). Consider the integrals
The integral I 1 can be computed as while The assertion follows. If (A + K, B + K) satisfies (QP) but not (P) or (E), let A + K = A 1 ∪ A 0 , B + K = B 1 ∪ B 0 be quasi-periodic decompositions of A + K and B + K with quasi-period W, such that A 0 +B 0 is a unique expression element of A +B +W +K in G/(W + K), and m G/K (A 0 + K + B 0 + K) = m G/K (A 0 + K) + m G/K (B 0 + K) Assume, without loss of generality, that A 1 = ∅. We will show that m(A 0 + K) > 0. Assume, to get a contradiction, that m(A 0 + K)     (2.14) If (2.13) holds for all b ∈ B such that A 0 +b ⊂ (A 1 +B) c , then m H(A+B) > 0, by Theorem 1.1, contradicting the hypothesis. Therefore, we assume (2.14) holds for some choice of b ∈ B and show that B = B 1 ∪ B 0 is a quasi-periodic decomposition. What remains to be shown is that Now (2.14) implies which is a disjoint union. It follows that A 0 + B 0 + H is a unique expression element of A + B + H in G/H, since otherwise there exists a ∈ A 1 and b ∈ B such that a + b + H ⊂ m (A 0 + B 0 + H) ∩ (A + B), contradicting the assumption that m (H(A + B)

Proof. (i) If
A is contained in a coset of H, we may apply Lemma 2.14 to conclude that B has a quasi-periodic decomposition with respect to some compact open H H. Otherwise, write C for (A+B) c , and note that the assumption that A+B is quasi-periodic with quasi-period H implies C ∼ C ⊂ C, where C is quasi-periodic with respect to H. Lemma 2.13 implies −B, C is an irreducible nonextendible sur-critical pair with m H(−B + C) = 0 and −B + C ∼ A c . Irreducibility of (−B, C) implies −B + C ∼ −B + C ∼ A c . Lemma 2.15 then implies that −B is quasi-periodic, and the same lemma applied to (A, B) gives a quasi-periodic decomposition of (A, B) satisfying (QP). This is the classical e-transform, whose properties are well-documented. In particular, A e + B e ⊂ A + B, and m(A e ) + m(B e ) = m(A) + m(B) whenever B e is nonempty. See [8] or [12] for further exposition.
Likewise, if a ∈ A ∩ B, then a + H is essentially contained in A. The claim follows.
Proof of claim. The definitions of A 0 and B 0 imply A + H contains B 0 , while A 1 + H and A 2 + H are disjoint from B 0 , and B 1 + H and We now consider two cases.
We assume also that for all a ∈ A 2 , a+B ⊂ A e + B e , or else we could repeat the argument of Case 1 with the roles of A and B reversed. Since (A, B) is nonextendible, we get that b + H ⊂ m B for all b ∈ B 2 , and a + H ⊂ m A for all a ∈ A 2 .

We now have
and B 0 ∩ A = ∅. Also, A 0 + H = B 0 + H. This situation is depicted in Figure 1: the vertical rectangles are cosets of H, while contra the conclusion, A 0 and B 0 are shown as occupying two cosets of H. Our immediate goal is to show that A + B has a quasi-periodic decomposition with quasi-period H.
If the pair (A (n) , B (n) ) is reducible, then n > 0, and Corollary 2.5 says that (A (n) , B (n) ) has a quasi-periodic decomposition. Since A + B ∼ A (n) + B (n) , Part If the construction of (A (n) , B (n) ) yields conclusion (ii) for some n, we are done. Otherwise, we have (i).
We now prove a special case of Theorem 1.4. Proof. The hypotheses of Lemma 3.1 are satisfied. If (ii) holds in the conclusion of Lemma 3.1, the conclusion of Theorem 1.4 is true. Now we must analyze the case where the conclusion (i) holds in Lemma 3.1. We follow the arguments of [8], §4.
For the following sequence of claims, fix an irreducible nonextendible sur-critical pair for G satisfying H(A + B) = {0}, and and fix a sequence (A (n) , B (n) ) of pairs satisfying (i) of Lemma 3.1. Proof of Claim 1. Suppose not. Then, since B (n+1) ⊂ B (n) for all n, there exists a neighborhood U of 0 and elements x n , y n ∈ B (n) such that x n − y n / ∈ U for all n ∈ N. Since G is compact, the sequences x n , y n have limit points x, y, with x − y = 0. Note that A ≤ a n · c/2, where the estimate m(B (n) + G ) ≤ a n is a consequence of the fact that χ(B (n) ) is contained in an interval of length a n . Now the inequality m(B (n) ) ≤ a n · c/2 for all n > N contradicts Claim 3.
We now complete the proof of Proposition 3.2 under the assumption that G is not totally disconnected and conclusion (i) holds in Lemma 3.1. By Claim 4, G is isomorphic to F × T for some finite group F. Let G 0 be the connected component of the identity in G. By Claim 1, we can choose n sufficiently large that B (n) is contained in a coset of G 0 . If A (n) is contained in a coset of G 0 , then A and B are each contained in a coset of G 0 . Then Theorem 1.2 implies the existence of a homomorphism χ : G 0 → T and intervals I, J ⊂ T such that A ⊂ a + χ −1 (I), B ⊂ b + χ −1 (J), for some a, b ∈ G, and m(A) = m(χ −1 (I)), m(B) = m(χ −1 (J)). Then we conclude (K) in Theorem 1.4. If A (n) is not contained in a coset of G 0 , Lemma 2.14 then yields a quasi-periodic decomposition of A (n) with respect to G 0 , and Lemmas 2.15 and 2.16 now imply that A (n) + B (n) , A (n) , and B (n) are quasiperiodic with quasi-period G 0 . Since A (n) + B (n) ∼ A + B, we apply Part (ii) of

Reduction from locally compact to compact
The structure theorem for locally compact abelian groups allows one to identify a sur-critical pair in an arbitrary locally compact abelian group G with a sur-critical pair in a compact quotient of G. We state a convenient version of the structure theorem for reference.  Proof. Assume, to get a contradiction, that the closure of A is not compact. It follows from Theorem 4.1 that there is a neighborhood U of 0 ∈ G and an infinite set A ⊂ A such that the sets a + U, a ∈ A are mutually disjoint. Now there is a b ∈ B so that m B ∩ (b + U ) > 0, and A + B contains all the mutually disjoint sets a + (B ∩ (b + U ) , where a ∈ A . Then m(A + B) = ∞. Proof. By Theorem 4.1, G has a compact open subgroup G 0 isomorphic to R n × K, where K is a compact abelian group, and the quotient G/G 0 is discrete. Since A and B have compact closures in G, the subgroup generated by (A+G 0 )∪(B +G 0 )∪ (A + B + G 0 ) in G/G 0 is finitely generated, with generators e 1 + G 0 , . . . , e d + G 0 . Thus we may assume G/G 0 is finitely generated. Let n 1 , . . . , n d be sufficiently large integers that the quotient map G → G/ n 1 e 1 , . . . , n d e d is one-to-one on A ∪ B ∪ (A + B). Now we may assume G/G 0 is finite, with generators f 1 , . . . , f d .

Acknowledgements
This work was conducted while the author was a postdoctoral fellow in the Department of Mathematics at the University of British Columbia. The author thanks Izabella Laba and Malabika Pramanik for financial support and helpful discussions. Discussions with Michael Björklund and Alexander Fish provided motivation for this work.