{'fraction_non_alphanumeric': 0.0822363021568563, 'fraction_numerical': 0.02130593217556876, 'mean_word_length': 3.1665982765695526, 'pattern_counts': {'":': 0, '<': 10, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the critical pair theory in abelian groups : Beyond Chowla's Theorem

We obtain critical pair theorems for subsets S and T of an abelian group such that |S+T|<|S|+|T|+1. We generalize some results of Chowla, Vosper, Kemperman and a more recent result due to Rodseth and one of the authors.

The Cauchy-Davenport Theorem was generalized to abelian groups by several authors including Mann [20] and Kneser [19]. The first generalization to cyclic groups is due to Chowla [3]: it states Theorem 1 Let S, T be nonempty subsets of Z/nZ such 0 ∈ S. Assume that every element of S \ {0} has order exactly n. Then |T + S| ≥ min(n, |S| + |T | − 1).
Subsets achieving equality in an additive theorem are known as critical pairs of the theorem. One may easily check that the only interesting critical pairs for the Cauchy-Davenport Theorem arise when |S|, |T | ≥ 2 and |S + T | ≤ p − 2. Under these assumptions Vosper's Theorem [24] states that |S + T | = |S| + |T | unless both S and T are arithmetic progressions with a common difference. This statement determines the critical pairs of the Cauchy-Davenport Theorem.
Generalizing Vosper's Theorem to arbitrary abelian groups requires a lot of care. The importance of this question was mentioned by Kneser in [19]. Motivated by Kneser's work, Kemperman proposed in [18] a recursive procedure which generalizes Vosper's Theorem to abelian groups. The main tools used by Kemperman are basic transformations introduced by Cauchy, Davenport and Dyson [21]. One of the results obtained by Kemperman is the following: Theorem 2 (Kemperman, [18]) Let G be an finite abelian group and let S, T be subsets of G such that |S| ≥ 2, |T | ≥ 2 and |S + T | = |S| + |T | − 1 ≤ p − 2, where p is the smallest prime divisor of |G|. Then S and T are arithmetic progressions with the same difference.
Note that the existence of a short direct proof for this result is unlikely since the statement contains Vosper's Theorem. This result has been recently extended to non abelian groups by Károlyi [17] and independently by one of the authors [8,Theorem 3.2].
By using the additive transformations mentioned above, Rødseth and one of the authors recently characterized the critical pairs of Vosper's Theorem [10] : Theorem 3 Let S, T be subsets of a group of prime order Z/pZ, with |T | ≥ 3 and |S| ≥ 4 such that |S + T | = |S| + |T | ≤ p − 4.
Then S and T are included in arithmetic progressions with the same difference and of respective lengths |S| + 1 and |T | + 1.
There are several methods currently available in additive theory. One of them is based on Fourier analysis. Examples of applications of this method can be found the monographs of Freiman [15] and Tao and Vu [23], or in the papers by Deshouillers and Freiman [5], and by Green and Ruzsa [6]. Another powerful tool is the polynomial method introduced by Alon, Nathanson and Ruzsa [1]. Károlyi recently [16] used this method to obtain a remarkable critical pair theorem for restricted sums.
In this paper we obtain improvements of some of the above results using the isoperimetric method. This method has been used to generalize addition theorems to non abelian groups in some papers including [25,12,8,11]. It also derives additive inequalities, mainly from the structure of the k-atoms of a set. If S is a generating subset containing 0 of an abelian group G, a set A is called a k-atom of S if it is of minimum cardinality among subsets X such that |X| ≥ k, |X + S| ≤ |G| − k, and |X + S| − |X| is of minimum possible cardinality (see Section 2 for detailed definitions). It is proved in [7] that any 1-atom containing 0 is a subgroup. This result implies easily Mann's generalization of the Cauchy-Davenport Theorem. The structure of 2-atoms has proved more difficult to describe but potentially gives stronger results: 2-atoms have been used in [9,13] to derive critical pair results. In groups of prime order, the description of 2-atoms was completed by two of the present authors in [22]. Atoms of higher order were used in [14] to classify sets S, T ⊂ Z/pZ with |S + T | ≤ |S| + |T | + 1.
In the present paper we first study the structure of 2-atoms in general abelian groups. Our main result in the first part of this paper is Theorem 21: broadly speaking it states that, under some technical conditions that will be shown to be quite tight, 2-atoms have cardinality 2 or are subgroups. In the rest of the paper we apply this fact to obtain critical pair results.
We shall first obtain a critical pair result for Chowla's Theorem 1 which reduces to Vosper's Theorem if n is a prime. To be precise, we will actually be dealing with a strengthened version of Theorem 1 (Corollary 8) that only requires the order of every element of S \ {0} to exceed |S| − 1 rather than to equal n. We call this requirement a weak Chowla condition. The description of the corresponding sets S and T are obtained in Theorem 14 and Corollary 16.
We then move on to give a description of subsets S, T , with |S + T | ≤ |S| + |T |, in arbitrary abelian groups provided S contains no element of order less than |S| + 1 (another weak Chowla condition). We show that, if the abelian group has no subgroups of order 2 or 3, then S and T are made up of arithmetic progressions with at most one missing element and periodic subsets with at most one missing element, see Theorems 28 and 29. This last result is a generalization to abelian groups of Theorem 3 of Rødseth and one of the authors, since it reduces to it when the group is of prime order.
The paper is organized as follows: Section 2 gives some preliminary results and Section 3 uses them to derive a solution to the critical pair problem for Chowla's Theorem and its strengthened version. Section 4 works out some tools necessary to Section 5 which is devoted to the description of 2-atoms. Sections 6 and 7 make up more preliminary material for section 8 which derives the generalization to abelian groups of Theorem 3.

Isoperimetric tools
In this section we recall known results on isoperimetric numbers of subsets in finite abelian groups and derive some consequences relevant to us later on. Our prime objects of concern are the 2-atoms of a subset: we shall see that they are either subgroups or Sidon sets and, in the last case, they have the largest possible isoperimetric numbers.
Let S be a subset of a finite abelian group such that 0 ∈ S. Denote by S the subgroup generated by S. For a positive integer k, we shall say that S is k-separable if there exists X ⊂ S such that |X| ≥ k and |X + S| ≤ | S | − k.
Suppose that S is k-separable. The k-th isoperimetric number of S is then defined by κ k (S) = min{|X + S| − |X| X ⊂ S , |X| ≥ k and |X + S| ≤ | S | − k}. (1) For a k-separable set S, a subset X achieving the above minimum is called a k-fragment of S. A k-fragment with minimal cardinality is called a k-atom.
The following easy facts will be used regularly throughout the paper: • The translate A + g of a k-atom A is also a k-atom.
If S is not k-separable, we shall put by convention κ k (S) = k|S| − 2k + 1 so as to have, for all |S| ≥ k, The definition of a k-atom implies the following lemma: Lemma 4 Let 0 ∈ S be a k-separable subset of a finite abelian group. Let A be a k-atom and suppose that |A| > k. Then, for each a ∈ A and s ∈ S we have Proof.
In other words, no element x in S + A can be uniquely written as x = s + a, s ∈ S and a ∈ A. This means that Next we recall: Let 0 ∈ S be a k-separable subset of a finite abelian group G. Let F be a k-fragment of S and g ∈ S . Then g − F and S \ (F + S) are k-fragments of −S. Moreover κ k (−S) = κ k (S).
The following is a particularly useful property of k-atoms.
Lemma 6 (The intersection property [8]) Let 0 ∈ S be a k-separable subset of a finite abelian group G. Let A be a k-atom of S. Let F be a k-fragment of S such that A ⊂ F . Then The intersection property implies easily the following description of 1-atoms.

Corollary 7 ([7])
Let 0 ∈ S = S be a subset of a finite abelian group G. Let A be a 1-atom of S such that 0 ∈ A. Then A is the subgroup generated by S ∩ A. In particular κ 1 (S) is a multiple of |A|.
From these early results we can derive the following generalization of Chowla's Theorem: Corollary 8 Let 0 ∈ S be a generating subset of a finite abelian group G such that the order of every element of S \ {0} is at least |S| − 1. Then κ 1 (S) = |S| − 1.
In particular, for every nonempty subset X ⊂ G, we have

Proof.
If S is not 1-separable, then by definition we have S = S and by the convention preceding (2) we have κ 1 (S) = |S| − 1. Suppose therefore that S is 1-separable. Let A be a 1-atom of S containing 0. By Corollary 7, A is the subgroup of G generated by S ∩ A and The last inequality in the statement is a direct consequence of the definition of κ 1 .
Recall that a subset X of an abelian group is a Sidon set if no two pairs of (not necessarily distinct) elements in X have the same sum. In particular |S ∩ (S + x)| ≤ 1 for each x.
Corollary 9 Let 0 ∈ S be a k-separable subset of a finite abelian group G. Let A be a k-atom of S such that 0 ∈ A, and suppose that p ≥ k where p is the smallest prime divisor of |G|. Then either A is a subgroup of G or |A∩(x+A)| ≤ k −1 for every x ∈ G, x = 0. In particular a 2-atom of a 2-separable set is either a subgroup or a Sidon set.

Proof.
Without loss of generality we may suppose S = G. The double inequality k ≤ |A ∩ (x + A)| < |A| is forbidden by Lemma 6 because x + A is also a k-atom of S. Suppose that there is x ∈ G, x = 0, such that A = A + x. Then we have A = A + x : hence A ∩ (a + A) ⊃ a + x for every a ∈ A. Since | x | ≥ p ≥ k, Lemma 6 implies that we have A = a + A for every a ∈ A and A is a subgroup.
Proof. Suppose on the contrary that κ 1 (S) ≤ |S| − 2. Then S is 1-separable. Let 0 ∈ A be a 1-atom of S. Then A is a nonnull subgroup of G and κ 1 (S) is a multiple of |A|.
The next result determines the second isoperimetric number of Sidon sets. In what follows we use the following notation. Given a subgroup H of G, by the decomposition of a subset S ⊂ G modulo H we mean the minimal partition of S into nonempty subsets, each one contained in a single coset of H.

Lemma 11
Let 0 ∈ S be a subset of a finite abelian group with |S| ≥ 3. If S is a Sidon set then κ 2 (S) = 2|S| − 3.
Proof. Let G = S . Suppose S is 2-separable, otherwise the result follows by the convention preceding (2).
If A is a Sidon set, then Suppose that A is a subgroup. Then κ 2 (S) is a multiple of |A|. In particular, |A| ≤ 2|S|−4. But then, since |A| ≥ 4, again a contradiction.
The following corollary is a result obtained in a more general context in [9]. The simple proof given here is similar to a proof given in [22].
Corollary 12 Let S be a generating set of the finite abelian group G with 0 ∈ S, |S| ≥ 3 and
If A generates G then 2|A|−3 = κ 2 (A) ≤ |S+A|−|S| = |A|+m, a contradiction. Therefore we may assume that A generates a proper subgroup Q of G. Let S = S 1 ∪ · · · ∪ S j , where j ≥ 2, be the decomposition of S modulo Q. We may assume that |S 1 + A| ≤ · · · ≤ |S j + A| and, by translating S, that 0 ∈ S 1 .
against our assumption. Therefore we may assume that It follows that |A| ≥ |Q| − 1, which is impossible since A is a Sidon set.
Finally, the following lemma will be useful to us in ruling out the possibility that a 2-atom is a subgroup.

Lemma 13
Let 0 ∈ S be a 2-separable subset of a finite abelian group G. Suppose A is a 2-atom of S which is a subgroup of cardinality at least 3. Then there exists s ∈ S, s = 0, such that the order of s is not more than κ 2 (S).

Proof.
Note that if A is a subgroup then κ 2 (S) is a multiple of |A|. By Lemma 4 we have Therefore there is a non-zero element s of S in A, and its order is not more than |A| ≤ κ 2 (S).

Critical pairs under the weak Chowla condition.
With the previous results we can already prove a critical pair theorem improving on the theorems of Chowla and Vosper. We first state its isoperimetric version. Recall that a subset S of an abelian group G is periodic if there is a nonnull subgroup H of G such that S +H = S. In other words, S is a union of cosets of H.

Theorem 14
Let 0 ∈ S be a generating 2-separable subset of a finite abelian group G such that κ 2 (S) ≤ |S| − 1. Also assume that every element of S \ {0} has order at least |S|. Then either S is an arithmetic progression or S \ {0} is periodic.

Proof.
By Corollary 8 we have κ 1 (S) = |S| − 1. Let 0 ∈ A be a 2-atom of S. Assume |S| ≥ 3 otherwise there is nothing to prove. By Lemma 13, the condition on the order of elements of S implies that A is not a subgroup. But then Corollary 12 implies that we have |A| = 2, say A = {0, r}. Assume first that r generates G. This forces S to be an arithmetic progression with difference r. Assume now that r generates a proper subgroup for all but one subscript. In particular S ∩ H = {0} since otherwise S contains a nonzero element with order at most |H| ≤ |S| − 1.
The above theorem will translate into a Chowla-type characterization of sets S and T with small sumset, this will be Corollary 16. The next result is a generalization of Theorem 2.
By the stabilizer of a subset X of an abelian group G, we mean the set of group elements x ∈ G such that X + x = X.
Proposition 15 Let 0 ∈ S be a generating subset of a finite abelian group G and let 0 ∈ T be a subset of G. Let Q denote the stabilizer of S \ {0}. Suppose that Also assume that every element of S * = S \ {0} has order ≥ |S|. Let σ : G → G/Q denote the canonical projection. One of the following holds: (ii) or σ(S) and σ(T ) are arithmetic progressions with the same difference. Moreover, at most one member of the decomposition of T modulo Q is not a complete coset modulo Q.
Proof. Either T = {0} and thus T ⊂ Q or the conditions on S imply that S is 2-separable and κ 2 (S) ≤ |S| − 1. Assume first Q = {0}. By Theorem 14, S is an arithmetic progression. It follows easily that T is an arithmetic progression with the same difference. Assume now Q = {0}.
We have |σ(T ) + σ(S)| ≤ |σ(T )| + |σ(S)| − 1. Otherwise there are |σ(S)| cosets in σ(T ) + σ(S) not present in σ(T ). But all these cosets are saturated in Moreover, the order of every element x ∈ σ(S) \ {0} is at least ⌈|S|/|Q|⌉ = |σ(S)|. Since the stabilizer of σ(S) * = σ(S * ) must be {0}, either σ(T ) = 0 and T ⊂ Q or Theorem 14 in G/Q implies that σ(S) is an arithmetic progression. It follows now that σ(T ) is an arithmetic progression with the same difference. Since σ(T ) contains at most a single element that is not expressible in G/Q in two different ways as a sum of one element of σ(S) and one element of σ(T ), we deduce that at most one coset modulo Q that intersects T is not included in T .

Corollary 16
Let 0 ∈ S and T be non-empty subsets of a finite abelian group G. Suppose that where Q denotes the stabilizer of S \ {0} and H is the subgroup of G generated by S. Also assume that every element of S \ {0} has order at least |S|. Let T 1 ∪ T 2 ∪ · · · ∪ T j be a decomposition of T modulo H such that Then |T i | = |H| for all i ≥ 2. Moreover one of the following conditions holds: (ii) σ(S) and σ(T 1 ) are arithmetic progressions with the same difference, where σ : G → G/Q denotes the canonical projection. Proof.
By Corollary 8 we have κ 1 (S) = |S| − 1. If j ≥ 2 we have |T 2 + S| = |H| since otherwise, By Proposition 15, either T 1 ⊂ Q or σ(S) and σ(T 1 ) are arithmetic progressions with the same difference. Now, if 0 ∈ T 1 then the same argument gives At the heart of the proof of Theorem 14 was the claim that, under the right conditions, a 2-atom containing the zero element is of cardinality 2 or is a subgroup. In Section 5 we shall find more general conditions under which we can make the same claim. Before that we need some more tools.

The fainting technique
In this section we use a method developed in [22]. The idea is to consider the sequence of subsets (S + A) \ S, (S + 2A) \ (S + A), · · · , (S + iA) \ (S + (i − 1)A), · · · and to claim that if A is a 2-atom of S of cardinality |A| > 2, then this sequence must decrease and faint, implying that S is a "large" subset of G.
Let X and Y be subsets of an abelian group G. For each integer i ≥ 0 we denote by In what follows we use the notation Y * = Y \{0}. We start with the two following lemmas.
Lemma 17 Let G be an abelian group and let X,

Proof.
Suppose that the statement holds for all i, r ≤ i ≤ j, for some j ≥ r, and let The result follows by induction.
Lemma 18 Let 0 ∈ S be a 2-separable subset of a finite abelian group G and let 0 ∈ A be a 2-atom of S with cardinality |A| ≥ 3 which is not a subgroup of G. Then, denoting

Proof.
Without loss of generality S generates G. The first part of the result is just Lemma 4. Now, since A is not a subgroup, we have S + A = S + A, otherwise we would have |S + A | − | A | < |S + A| − |A| in contradiction with A being a 2-atom. Therefore there exists x ∈ N 2 (S, A) = (S + 2A) \ (S + A). Recall that, by Lemma 5, the subset x − A is a 2-atom of−S and G \ (S + A) is a 2-fragment. Observe that x ∈ (x − A) ∩ (G \ (S + A)) and that x ∈ N 2 (S, A) means x − A is not contained in G \ (S + A): the intersection property of 2-atoms (Lemma 6) implies therefore that (x − A) ∩ (G \ (S + A) = {x}), but this means The following Lemma is a key tool for the proof of the main result of the next section. It says that, under some conditions, a set X verifying the statement of Lemma 18 with some other set must be a large subset of the ground group.
Lemma 19 (The Fainting Lemma) Let G be a finite abelian group and let X, Y ⊂ G with 0 ∈ X ∩ Y and set m = |X + Y | − |X| − |Y |. Assume that Y generates G and that Then |X| ≥ |G| − m + 4 2 .

Proof.
Since X + Y = X + (Y \ {y}) for any y ∈ Y , we have X + Y = X + Y * and X + (Y − y) = X + (Y * − y). By induction on i it is seen that Let H be the subgroup of G generated by Y * − y. One can verify easily that H = n(Y * − y), and hence By Lemma 17, Since is a union of cosets of the subgroup H generated by Y * − y. In particular H = G and, by (4), y)) + y contains a full coset of this subgroup. However, we have N i (X, Y ) ∩ X = ∅ and, by (3), X + H = G, a contradiction. Let ℓ be the largest integer for which N ℓ (X, Y ) = ∅. We have just shown that, for each i, 1 ≤ i < ℓ, Therefore, Since |N 1 (X, Y )| = |Y | + m we have |N 2 (X, Y )| ≤ m + 2. Hence, since 3 ≤ |Y | ≤ m + 3, the largest possible value in the right hand side of inequality (6) is taken if |Y | = 3 and ℓ = m + 3 giving as claimed.
We finish this set of preliminary results with the following Lemma.
Lemma 20 Let A and S be subsets of a finite abelian group Q. Assume that |A| = 3 and that for each a ∈ A we have S + A = S + (A \ {a}). Then 3|S| ≥ 2|S + A|.

Description of 2-atoms
The next theorem gives the structure of the 2-atoms for not too large subsets of an abelian group.
Theorem 21 Let G be a finite abelian group and let 0 ∈ S be a generating 2-separable subset of G such that |S| ≥ 3 and Let A be a 2-atom of S containing 0. If |S| < |G| − m+4 2 then either |A| = 2 or A is a subgroup of G.
Proof. Suppose that the conclusion of the theorem does not hold, so that 0 ∈ A is a 2-atom of S with |A| ≥ 3 which is not a subgroup. Then it follows from Corollary 9 that S is a Sidon set and then, by Lemma 11, κ 2 (S) = 2|S| − 3 ≥ |S|. In particular m ≥ 0.
By Corollary 12 we have |A| ≤ m + 3. Moreover, A * − a is also a Sidon set, and Lemma 10 implies that A satisfies condition (i) of the Fainting Lemma. By Lemma 18, S and A satisfy condition (ii) of the Fainting Lemma: therefore if A generates G its conclusion must hold. In that case we have |S| ≥ |G| − m+4 2 against the hypothesis of the Theorem. Therefore A must generate a proper subgroup Q of G. Let S = S 1 ∪ S 2 ∪ · · · ∪ S t be the decomposition of Since |S + A| = |S| + |A| + m, the decomposition of S + A modulo Q gives Now, as mentioned above, A is a Sidon set and by Lemma 10, we have κ 1 (A) = |A| − 1. Therefore |S i + A| − |S i | ≥ |A| − 1 for i ∈ V . Notice furthermore that Lemma 4 implies S i + A = S i + A * , so that |S i | ≥ 2 for each i ∈ I. Therefore, since by Lemma 11 we have κ 2 (A) = 2|A| − 3, we have |S i + A| − |S i | ≥ 2|A| − 3 for i ∈ W . Inequality (7) gives us In particular we have w ≤ 2 and v ≤ 3. Now for any i ∈ I let us write and, for J ⊂ I, put δ(J) = i∈J δ(i). Notice that δ(I) = |S + Q| − |S + A|, δ(U ) = 0, δ(V ) = v and that we have shown that m i ≥ |A| − 3 ≥ 0 for i ∈ W . We consider two cases.
If w = 1, say W = {1}, then (11) translates to If m = m 1 then we must have δ(V ) = 0 and the right hand side of (12) equals 1 + m 1 +4 2 . If m 1 < m then δ(V ) = v ≤ 3 and m 1 ≥ 0 imply that the right hand side of(12) is again ≥ 1 + m 1 +4 2 . In both cases this contradicts the Fainting Lemma applied to S 1 and A.
The following example shows that the result of Theorem 21 does not hold anymore if m = 5.

Example.
Take G = Z/7Z × Z/qZ where q > 7 is a prime. Consider the sets • S = {0, 1, 2, 4} × X where |X| = 4 and X is a Sidon set in Z/qZ and Then |S +A| = |S|+|A|+5. The group G has only two proper subgroups H 1 = Z/7Z×{0} and H 2 = {0} × Z/qZ and On the other hand, if B = {0, x} we have The last inequality being because, for any y = 0, |X ∪ (X + y)| ≥ 7 in Z/qZ since X is a Sidon set. Therefore subgroups and subsets of size 2 are not 2-atoms of S. Furthermore we have κ 2 (S) ≥ |S| + 5 since otherwise Theorem 21 would apply: therefore A is a 2-atom of S and κ 2 (S) = |S| + 5.
Finally, note that Theorem 21 together with Lemma 13 give a sufficient condition to rule out the possibility of a 2-atom being a subgroup.

Corollary 22
Let G be a finite abelian group and let 0 ∈ S be a generating 2-separable subset of G such that |S| ≥ 3 and Also assume that every non zero element of S has order at least |S| + m + 1.

Atoms of small sets
We next show some results about k-atoms of small sets.

Lemma 23
Let S be a 4-separable generating subset of a finite abelian group such that 0 ∈ S and κ 4 (S) = |S| = 3. Let 0 ∈ A be a 4-atom of S. Then |A| = 4.

Proof.
Let G = S . Suppose that |A| > 4. We shall apply the Fainting Lemma to A and S.
Take z ∈ S * . By Lemma 4 we have By Lemma 17 we have N 2 (A, S) − S * ⊂ N 1 (A, S). Now we may apply the Fainting Lemma and obtain |A| ≥ |G|−6. But then |A+S| ≥ |G|−3 contradicting that A is a 4-fragment of S.
Claim. S * is an arithmetic progression.
Let us write N i = N i (A, S), i ≥ 0. Note that |N 1 | = |S + A| − |A| = κ 3 (S) = |S| = 4. For each subset X ⊂ S and for each i ≥ 1, let us denote by N X i the set of elements u ∈ N i such that u − X ⊂ N i−1 and X is a maximal subset of S with this property. By the definition, We have just shown that: By Lemma 4, for each x ∈ S * , we have On the other hand, for each x ∈ S * , inequality (13) implies Let us now estimate |N X 2 | and |N X 3 | for X ⊂ S * . Note that by Corollary 8 as κ 1 (Z) = |Z| − 1 for each subset 0 ∈ Z ⊂ G with |Z| ≤ 3, since the order of any nonzero element in G is at least 5. Therefore, using (14), we have Since there are at most two 2-subsets of S * for which |N X 1 | = 2, we have Since |N 1 | = 4, then N S * 2 − S * cannot be a coset. Therefore, since This implies that |N S * 2 | ≤ 2. Suppose that |N S * 2 | ≤ 1. Then |N 2 | = X⊂S * |N X 2 | ≤ 3 and, by applying (14) with i = 3 and 4, we get |N 3 | = |N S * 3 | ≤ 1 and |N 4 | = 0. Therefore |N 2 | + |N 3 | ≤ 4 < |G| − |S + A|. This means that Y = A ∪ N 1 ∪ N 2 ∪ N 3 = G and Y + S = Y , which contradicts that S generates G.
Suppose now that |N S * 2 | = 2. Then |N S * 2 − S * | = |S * | + 1 which implies that S * is an arithmetic progression. This proves the claim. Now we have S = {0, a, a + d, a + 2d} for some d ∈ G. By repeating the argument of the claim to S − a − d we get that {−a − d, −d, d} is an arithmetic progression as well.

Quasi-progressions
A subset S of an abelian group G will be called a quasi-progression of difference r if S is not a progression with difference r and if S can be obtained by deleting an element of an arithmetic progression of difference r.

Lemma 25
Let 0 ∈ S be a quasi-progression with difference r in the cyclic group Z/nZ. Suppose that S generates Z/nZ and |S| ≥ 3. Let T ⊂ Z/nZ be such that |T | ≥ 3 and Then one of the following conditions holds: (i) T is either a quasi-progression with difference r or a progression with difference r.
(ii) n = 12 and T is a coset of order 4.
For a subset X ⊂ Z/nZ let us call connected components of X the maximal arithmetic progressions with difference 1 contained in X. Case 1. There is a connected component C 1 of T = Z/nZ \ T such that |C 1 | ≥ |S|.
If U = {u, u + 1, · · · , v} is a connected component of T , then v + 1 ∈ (S + T ) \ T . Since |S + T | ≤ |S| + 4, it follows that T has exactly 4 components, and for each such component Lemma 26 Let S and T be subsets of Z such that |S| = 3, |T | = 4 and |S + T | = 7. Then S is either a progression or a quasi-progression.
The proof is an easy exercise.
Lemma 27 Let S be a 4-separable generating subset of an abelian group G of order n such that 0 ∈ S, |S| = 3 and κ 4 (S) = |S| = 3. Assume moreover that gcd(n, 6) = 1. Then G is a cyclic group and S is a quasi-progression.
We show first that every element of S \ {0} generates G. Suppose on the contrary that x generates a proper subgroup K of G. Since gcd(|G|, 6) = 1 we have min{|H|, |G/K|} ≥ 5.
Let φ denote the canonical morphism from G onto G/K. Decompose A = A 1 ∪ · · · ∪ A j , j ≥ 2, modulo the subgroup K and assume that 0 ∈ A 1 and |A 1 | ≤ |A i |, i ≥ 2. Notice that Since |A| = 4 and gcd(|G|, 6) = 1, we have |A + {x, y}| ≥ |A| + 1. Assume first that |A + {x, y}| = |A| + 1. Then A is an arithmetic progression with difference y − x. But 0 ∈ A and hence y − x is invertible since A generates G. Without loss of generality we may assume A = {0, 1, 2, 3}. Now it comes easily that S is a quasi-progression, and the result holds.
Now since x is invertible in G = Z/nZ, we may write, without loss of generality, S = {0, 1, t} with |A ∩ (A + 1)| ≥ 2. By translating and multiplying by −1, we can also assume that t ≤ (n + 1)/2 (notice that n 2 is not a unit if n is even). Therefore A can be represented by two pairs of consecutive integers, and hence by a subset of 4 integers included in an interval of length ≤ (n + 1)/2. On the other hand, one of the following two possibilities holds for S: • S can be represented by a subset of an integral interval of length ≤ (n − 3)/2. In that case the sum A + S in Z/nZ has the same cardinality as the sum A + S in Z, and we are done by Lemma 26.
• We have t = (n−1)/2, in which case S is included in an arithmetic progression of length 4 and difference 2 −1 (2 is invertible since n is odd) and we are done.

Improving both the Theorems of Chowla and of Vosper
Next we shall generalize Theorem 14 to the case when |S + T | ≤ |S| + |T |. Our result is also a generalization to abelian groups of Theorem 3, i.e. the main result of [10]. Let us state it first under an isoperimetric formulation. Let us call a set quasi-periodic if it can be obtained by deleting one element from a periodic set.
If every element of S \ {0} has order at least |S| + 1, then either S is a quasi-progression or S \ {0} is quasi-periodic.
Claim. The result holds if A generates a proper subgroup K of G.
Assume first that A = K. In this case κ 3 (S) is a multiple of |A| and hence |S| ≥ |A|. It follows that S ∩ A = {0}, since otherwise A would contain an element of order at least |S| + 1. Now S + A is the disjoint union A ∪ (S * + A). Hence |S * + A| = |S * | + 1 so that S * is quasi-periodic and the result holds.
We may therefore assume that A generates G. We now consider three cases.
In that case A is 4-separable and κ 4 (A) ≤ |A|. If κ 4 (A) < 3, then Theorem 14 implies that A is a progression and thus S is a quasi-progression. If κ 4 (A) = 3, then by Lemma 27, A is a quasi-progression. By Lemma 25, S is a quasi-progression.
Theorem 28 translates into a characterization of subsets S and T such that |S + T | ≤ |S| + |T | under some Chowla-type conditions. This was our final goal in this paper.
Let 0 ∈ S be a generating subset of G such that |S| ≥ 4 and every element in S * has order at least |S| + 1. Let Q be a maximal subgroup such that |S * + Q| − |S * | ≤ 1 and let σ : G → G/Q denotes the canonical projection.
Let T be a subset of G such that |T | ≥ 3 and suppose that |S + T | = |S| + |T | ≤ |G| − 4. Then the following holds: • If Q = {0} then S and T are progressions or quasi-progressions with the same difference.
The conditions |S| + |T | = |S + T | ≤ |G| − 4 and |T | ≥ 3 imply that S is 3-separable and that κ 3 (S) ≤ |S|. By Theorems 14 and 28, S is an arithmetic progression or quasi-progression. By Lemma 25 it follows that T is an arithmetic progression or quasi-progression with the same difference.
This holds clearly if S * is Q-periodic. So we may assume |S * + Q| − |S * | = 1. Let us then denote by S 1 the unique subset of S of size |Q| − 1 in the decomposition of S modulo Q. If Σ is not Q-periodic then some Q-coset must have a trace U of size |Q| − 1 on the set Σ, and we have U = S 1 + T ′ where T ′ = (a + Q) ∩ T , for some a. Since |S 1 | = |Q| − 1 we must have |T ′ | = 1. Note also that σ(S 1 ) + σ(T ′ ) cannot be obtained in any other way as a sum of an element of σ(S) and of an element of σ(T ), therefore (S 1 + T ′ ) ∩ (S + (T \ T ′ )) = ∅, hence |S + (T \ T ′ )| < |S| + |T \ T ′ | − 1, but this contradicts κ 1 (S) = |S| − 1 (Corollary 8) and proves (19).
By our assumptions, Q is a maximal subgroup such that |S * + Q| − |S * | ≤ 1. This is easily seen to imply that σ(S * ) is not periodic. Moreover, each element in σ(S) * has order at least (|S| + 1)/q ≥ |σ(S)| − 1 + 1/q. Then, by Proposition 15, σ(S) and σ(T ) are arithmetic progressions with the same common difference d. Since −d is also a difference of σ(S) and σ(T ), we may assume without loss of generality that the terminal element u of σ(S) is not 0. Therefore if we set S ′ = σ −1 (u) ∩ S we have |S ′ | ≥ |Q| − 1. Let us suppose, without loss of generality, that the initial element of σ(T ) is 0.

Remark.
One may wonder what happens if we remove from Theorem 29 the hypothesis |σ(S + T )| < |G|/|Q| − 1. Then the sets σ(S) and σ(T ) are not necessarily arithmetic progressions any more. However, one may show that there again exists T 1 ⊂ T , such that |σ(T 1 )| ≤ 1 and T \ T 1 is Q-periodic or Q-quasi-periodic. We leave out the details.