OOn Sums of Generating Sets in Z n ∗ October 10, 2018
Abstract
Let A and B be two affinely generating sets of Z n . As usual, we denote their Minkowski sumby A + B . How small can A + B be, given the cardinalities of A and B ? We give a tight answer tothis question. Our bound is attained when both A and B are unions of cosets of a certain subgroupof Z n . These cosets are arranged as Hamming balls, the smaller of which has radius 1.By similar methods, we re-prove the Freiman–Ruzsa theorem in Z n , with an optimal upperbound. Denote by F ( K ) the maximal spanning constant |(cid:104) A (cid:105)| / | A | over all subsets A ⊆ Z n withdoubling constant | A + A | / | A | ≤ K . We explicitly calculate F ( K ), and in particular show that4 K / K ≤ F ( K ) · (1 + o (1)) ≤ K / K . This improves the estimate F ( K ) = poly ( K )4 K , foundrecently by Green and Tao [17] and by Konyagin [23]. Much work has been devoted to the study of Minkowski sums of sets. Questions concerning suchsums come up in geometry, and are at the core of additive combinatorics. Research in this areahas blossomed in recent years, and even Tao and Vu’s monograph [36] no longer covers all the mostrecent developments. In this paper we concentrate on the Minkowski sum of two generating setsof Z n .We first review some of the relevant literature. Let G be an abelian group, and let A and B betwo finite subsets of G . As usual, we denote A + B = { a + b | a ∈ A, b ∈ B } and we ask about the minimum of | A + B | , given the cardinalities of A and B .In general, the answer ranges from max( | A | , | B | ) to | A | + | B | −
1, depending on the structureof G . For a torsion-free G , if A and B are arithmetic progressions with the same step, then | A + B | = | A | + | B | −
1, which is optimal. Likewise, if G = Z p is cyclic of prime order, then theanswer is given by the Cauchy–Davenport theorem, | A + B | ≥ min( | A | + | B |− , | G | ) [2, 4]. Moreover,by a theorem of Vosper [38], if | A | + | B | < | G | then equality holds only for arithmetic progressions. Inthe other extreme case, G has a finite subgroup of a suitable cardinality. Thus, if H (cid:47)G is a subgroupof cardinality | H | = max ( | A | , | B | ), an optimal choice is to have A and B be subsets of H , in whichcase | A + B | = max ( | A | , | B | ). More generally, | A + B | can be as small as max ( | A | , | B | ) if and onlyif min( | A | , | B | ) ≤ | H | and | H | divides max( | A | , | B | ) [36, p. 55]. In the general case [8, 10], thesmallest possible cardinality of | A + B | is min ( (cid:100)| A | / | H |(cid:101) + (cid:100)| B | / | H |(cid:101) − | H | , where the minimumis over all finite subgroups H of G . In a sense, this result interpolates between the two extremes.In an optimal construction [1, 10] the sets A and B are contained in (cid:100)| A | / | H |(cid:101) and (cid:100)| B | / | H |(cid:101) cosetsof H , whose arrangement is a lexicographical variant of an arithmetic progression. In particular,for G a 2-torsion group this reduces to the well-studied Hopf–Stiefel function [20, 35, 39, 1, 7, 9]. Stability is a recurring theme in modern extremal combinatorics. Once an extremal problem issolved, it is interesting to explore what happens when we consider candidate solutions that do not ∗ Einstein Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel, [email protected] paper is based on the author’s MSc thesis, under the supervision of Prof. Nati Linial. a r X i v : . [ m a t h . C O ] J u l esemble the global optimum. The crucial feature of the above-mentioned optimal constructions isthat A and B are densely packed in cosets of properly chosen subgroups of G . We therefore returnto the original question, under the requirement that A and B are not allowed to be contained in aproper subgroup of G or a coset thereof. The affine span of A , denoted (cid:104) A (cid:105) , is the smallest coset (ofany subgroup) containing A . We say that A affinely generates G if (cid:104) A (cid:105) = G . Clearly this definitioncoincides with the usual notion of a generating set if 0 ∈ A . The refined problem is as follows: Ina finitely generated abelian group G , find min | A + B | as a function of | A | and | B | , where A and B are finite affinely generating subsets of G .Naturally, the structural properties of G play a role in this problem as well. For the torsion-freecase, G = Z d , this question and similar ones were discussed by Ruzsa [30], and a full answer wasfinally given by Gardner and Gronchi [15]. In the extremal construction, the smaller set is a simplexof d + 1 points, on one of whose edges lies an arithmetic progression, and the other set is roughlythe sum of several copies of it. As discussed there, this is analogous to the Brunn–Minkowskitheorem [33].Here we present the following lower bound for the opposite extreme of a 2-torsion group, G = Z n . Theorem 1.
Suppose
A, B ⊆ G = Z n such that (cid:104) A (cid:105) = G , B (cid:54) = ∅ and | A | ≤ | G | .If t is the largest positive integer such that | A | ≤ t + 12 t · | G | and ≤ k < t and w ∈ [ − , are such that | B | = (cid:0) t (cid:1) + (cid:0) t (cid:1) + ... + (cid:0) tk (cid:1) + w (cid:0) t − k (cid:1) t · | G | then | A + B | ≥ (cid:0) t (cid:1) + (cid:0) t (cid:1) + ... + (cid:0) tk (cid:1) + (cid:0) tk +1 (cid:1) + w (cid:0) t − k +1 (cid:1) t · | G | This bound is tight when w = 0 , and it is attained by the sets A = D t × Z n − t B = D tk × Z n − t A + B = D tk +1 × Z n − t where D tk = { x ∈ Z t | { i | x i = 1 } ≤ k } is a Hamming ball of radius k in Z t . The Freiman–Ruzsa theorem [31] is a major result in additive combinatorics. In the contextof the above discussion, it addresses the special case A = B . It states that if A is a subset of an r -torsion abelian group with | A + A | ≤ K | A | , then A is contained in a coset of cardinality at most F ( K ) | A | , with F ( K ) = K r K . The special case r = 2 has received considerable attention [5, 6,16, 17, 19, 23, 26, 32, 37]. Among the most recent contributions is work by Green and Tao [17] withfurther improvement by Konyagin [23]. It shows that one can take F ( K ) = 2 K + O (log K ) . Here weexactly determine the lowest possible value of F ( K ) for r = 2. Theorem 2.
For K ≥ , denote by t ≥ the unique integer for which: (cid:0) t (cid:1) + t + 1 t + 1 ≤ K < (cid:0) t +12 (cid:1) + ( t + 1) + 1( t + 1) + 1 For A ⊆ Z n such that | A + A | / | A | ≤ K , we have |(cid:104) A (cid:105)| / | A | ≤ F ( K ) where: F ( K ) = t ( t ) + t +1 · K ( t ) + t +1 t +1 ≤ K < t + t +12 t t +1 t + t +1 · K t + t +12 t ≤ K < ( t +12 ) +( t +1)+1( t +1)+1 This choice of F ( K ) is tight, and grows as Θ(2 K /K ) .Compression is an important tool from extremal set theory. Much progress in the application ofcompression to additive problems was made by Bollob´as and Leader in [1], and it is a key ingredientin Green and Tao’s proof in [17]. There is a whole range of compression operators C , that transform n arbitrary set A to another set C ( A ), with | C ( A ) | = | A | and | C ( A ) + C ( A ) | ≤ | A + A | . By afinite sequence of such compressions, it is possible to reduce to the case where A is compressed insome appropriate sense, and hence has certain structural properties, which make A + A easier tostudy. The difficulty is that C ( A ) need not be affinely generating even if A is. Green and Taohandled this difficulty by restricting the types of compression operators they used. Our approachis different. We employ more types of compression operators and we proceed as long as possiblewithout jeopardizing affine generation, i.e., as long as (cid:104) C ( A ) (cid:105) = (cid:104) A (cid:105) . Isoperimetric inequalities play an important role in our work. In our investigations of A + B ,we prove a new variant of the isoperimetric inequality for the hypercube. Overview:
In Section 2 we discuss compressions and other useful tools. We explore the keynotion of compression that maintains affine generation. In Section 3 Theorem 2 is proved, first inan asymptotic form, then with the exact expression. In Section 4 we establish Theorem 1. Theproof utilizes our new isoperimetric inequality.
In this section we briefly survey several concepts and results that are used below. These includethe lexicographic order and the Hopf–Stiefel function. Then we discuss compressions in Z n , in linewith Section 2 of [17], and we introduce the study of compressions that preserve affine generation. Throughout, we use the linear basis { e , e , ..., e n } for Z n . Elements x ∈ Z n are expressed as x = (cid:80) ni =1 x i e i . The correspondence between vectors x ∈ Z n and their supports { j | x j = 1 } ⊆{ , ..., n } = [ n ], is used to simplify certain notation and arguments.The lexicographic order is a total order on Z n . For x, y ∈ Z n we say that x ≺ y , if x i < y i forthe largest coordinate i for which x i (cid:54) = y i . For example, the ordering of Z is:0 ≺ e ≺ e ≺ e + e ≺ e ≺ e + e ≺ e + e ≺ e + e + e The height , (cid:126) ( x ) of an element x in a finite totally ordered set is x ’s place in that order. For aset of elements A we denote (cid:126) ( A ) = (cid:80) x ∈ A (cid:126) ( x ).If T ⊆ Z n , then its initial segment of size a , denoted IS ( a, T ), is the set of the a smallestelements of T in the lexicographic order. We use the abbreviation IS ( a ) = IS ( a, Z n ) for n ∈ N large enough. For the reader’s convenience we prove the following observation of Bollob´as and Leader [1].
Proposition 3.
For two initial segments IS ( a ) , IS ( b ) ⊆ Z n , the sum IS ( a ) + IS ( b ) is an initialsegment as well.Proof. For z ≺ x + y , we claim that z = x (cid:48) + y (cid:48) for some x (cid:48) (cid:22) x and y (cid:48) (cid:22) y . Let i ∈ N be largestindex such that x i = 1 or y i = 1. Say x i = 1. If z i = 0, then clearly z ≺ x , so we can take x (cid:48) = z and y (cid:48) = 0. If z i = 1, then note that ( z − e i ) ≺ ( x − e i ) + y . By induction on i , obtain( z − e i ) = x (cid:48)(cid:48) + y (cid:48)(cid:48) for x (cid:48)(cid:48) (cid:22) ( x − e i ) and y (cid:48)(cid:48) (cid:22) y , and choose x (cid:48) = x (cid:48)(cid:48) + e i and y (cid:48) = y (cid:48)(cid:48) .The Hopf–Stiefel binary function a ◦ b can be defined on N × N as follows: a ◦ b = | IS ( a ) + IS ( b ) | Proposition 3 can be restated as: IS ( a ) + IS ( b ) = IS ( a ◦ b ). This definition is relevant for us forthe following reason. The cardinality of a sumset of two sets of given cardinalities is minimized bytaking the two sets to be initial segments: a ◦ b = min (cid:26) | A + B | (cid:12)(cid:12)(cid:12)(cid:12) A, B ∈ Z n , | A | = a, | B | = b (cid:27) ote that here the sets are not required to be affinely generating. This result can be deduced bythe technique of compressions as we discuss below. See Lemma 6.In particular, taking A = IS ( a ) and B = IS ( b ) ∪ ( e n + IS ( b )) for n large enough, one canverify the sub-distributive law: a ◦ ( b + b ) ≤ a ◦ b + a ◦ b Similarly, one can deduce the recursive relations for a, b ≤ n : a ◦ (2 n + b ) = 2 n + a ◦ b (2 n + a ) ◦ (2 n + b ) = 2 n +1 These two formulas can be taken as an alternative definition of the Hopf–Stiefel function [27].The function first arose in works of Hopf [20] and Stiefel [35]. They used tools from algebraictopology to prove that a ◦ b provides a lower bound for solutions of the Hurwitz problem, concerningreal quadratic forms (see [34]). The relation to set addition in Z n was given by Yuzvinsky [39].As it turns out, the Hopf–Stiefel function arises in the study of several more problems in variouscontexts. There is also a base- p analog of the this function for p >
2, see [7]. For a survey, see [9].
For I = { i , i , ... } ⊆ [ n ], denote H I = (cid:104) , e i , e i , ... (cid:105) (cid:47) Z n . As usual, if H is a subgroup of G ,we denote by G/H the collection of all H -cosets in G . The I -compression of a subset A ⊆ Z n isdefined by: C I ( A ) = (cid:91) T ∈ Z n /H I IS ( | A ∩ T | , T )In words, in every H I -coset T we replace the elements of A ∩ T by a same-cardinality initial segment,with respect to the lexicographic order. We say A is compressed with respect to I , or I -compressed ,if C I ( A ) = A . In particular, lexicographic initial segments of Z n are exactly all [ n ]-compressed sets. Example 4. C { , , } ( { , e , e , e , e } ) = { , e , e , e + e , e } . This notion of compression is closely related to the operation bearing the same name fromextremal set theory (see, e.g., [12]). A subset of Z n naturally corresponds to a family, a.k.a. set-system, F of subsets of [ n ]. We freely move between these terminologies if no confusion can occur.An { i } -compression corresponds to the push-down operator T i , which replaces J ∈ F by J \ { i } provided that J \ { i } (cid:54)∈ F . If F is { i } -compressed for each i , then it is closed under taking subsetsand is called a downset . The shift operator S ij replaces j by i wherever possible. Namely, for every J with i, j (cid:54)∈ J it replaces J ∪ { j } by J ∪ { i } given that the former belongs to F and the latterdoesn’t. We say that F is shift-minimal if it is invariant to all shifts S ij where i < j . One cancheck that being { i, j } -compressed for all i, j ∈ [ n ] corresponds to being a shift-minimal downset.Compression can simplify matters substantially, while preserving several useful features of theset-system. Here are some observations about compressions. These and others are found in [17].The proofs are straightforward, working coset by coset. Lemma 5 (Properties of compressions) . Suppose A ⊆ Z n and I ⊆ [ n ] .(1) | C I ( A ) | = | A | .(2) C I ( A ) is I -compressed.(3) (cid:126) ( C I ( A )) ≤ (cid:126) ( A ) with equality iff A is I -compressed.(4) An I -compressed set is J -compressed for all J ⊆ I .(5) C I ( A ) ⊆ C I ( B ) for all A ⊆ B . Compressions behave well on sumsets. By Proposition 3, one can deduce that the sum oftwo I -compressed subsets is I -compressed too. The following well-known lemma deals with thecompression of a sum of two general subsets. For the sake of completeness, we prove it here,following [1] and [17]. emma 6 (Sumset compression) . Suppose
A, B ⊆ Z n and I ⊆ [ n ] . Then C I ( A ) + C I ( B ) ⊆ C I ( A + B ) . Consequently | C I ( A ) + C I ( B ) | ≤ | A + B | .Proof. We use a double induction, on | I | and on (cid:126) ( A ) + (cid:126) ( B ). For the induction step, suppose thatfor some J (cid:40) I either A or B is not J -compressed. In this case C I ( A ) + C I ( B ) = C I ( C J ( A )) + C I ( C J ( B )) ⊆ C I ( C J ( A ) + C J ( B )) ⊆ C I ( C J ( A + B )) = C I ( A + B )Both inclusions are by the induction hypothesis: the first one since (cid:126) ( C J ( A )) + (cid:126) ( C J ( B )) < (cid:126) ( A ) + (cid:126) ( B ) by property (3) of Lemma 5, and the second one since | J | < | I | and by property (5). Theequalities are by property (4).It only remains to verify the lemma for A and B that are both J -compressed for all J (cid:40) I . Westart with the simpler case n = | I | .What are the subsets of G = Z n that are J -compressed for all J (cid:40) [ n ]? By property (4), allinitial segments are such. If S ⊆ G is not an initial segment, then necessarily x / ∈ S and y ∈ S forsome consecutive x ≺ y . The only consecutive pair in G that is not contained in a proper H J -cosetis ( e + ... + e n − ) ≺ e n . One can verify, for example by S being [2 ...n ]-compressed, that the onlysuch set is S = H [ n − \ { e + ... + e n − } ∪ { e n } . In conclusion, it is enough to check the case where A and B are initial segments or equal to S . Now there are four cases to consider:1. If both A and B are initial segments, then by Proposition 3 A + B is an initial segment too. ⇒ C I ( A ) + C I ( B ) = A + B = C I ( A + B )2. If A = B = S , then note that | S | ≤ | S + S | and C I ( S ) = H [ n − . ⇒ C I ( S ) + C I ( S ) = C I ( S ) ⊆ C I ( S + S )3. If B = S and A is an initial segment with | A | ≤ | S | , then A = C I ( A ) ⊆ C I ( S ) = H [ n − . ⇒ C I ( A ) + C I ( S ) = C I ( S ) ⊆ C I ( A + S )4. If B = S and A is an initial segment with | A | > | S | then | A | + | S | > | G | .This means A + S = G , as the reader may verify by a standard pigeonhole argument. ⇒ C I ( A ) + C I ( S ) = G = C I ( G ) = C I ( A + S )The case n > | I | is implied by the case n = | I | : C I ( A ) + C I ( B ) = (cid:91) H c ∈ G/H I (( C I ( A ) + C I ( B )) ∩ H c )= (cid:91) H c ∈ G/H I (cid:91) H a + H b = H c (( C I ( A ) ∩ H a ) + ( C I ( B ) ∩ H b ))= (cid:91) H c ∈ G/H I (cid:91) H a + H b = H c ( C I ( A ∩ H a ) + C I ( B ∩ H b )) ⊆ (cid:91) H c ∈ G/H I (cid:91) H a + H b = H c C I (( A ∩ H a ) + ( B ∩ H b )) ⊆ (cid:91) H c ∈ G/H I C I (cid:32) (cid:91) H a + H b = H c (( A ∩ H a ) + ( B ∩ H b )) (cid:33) = (cid:91) H c ∈ G/H I C I (( A + B ) ∩ H c )= (cid:91) H c ∈ G/H I ( C I ( A + B ) ∩ H c )= C I ( A + B ) . The first and second inequalities are simply dividing into cases, according to the involved H I -cosets. The third one holds because compressions work coset-wise. Then there is inclusion by theassumption on the case I = [ n ], applied to our H I and translated to the relevant H I -cosets. Andthen, inclusion of initial segments, because the union is at least as large as each of its components.The three remaining equalities are similar to the first three. .4 Compressions that Preserve Affine Generation As Lemma 6 shows, in the problems we consider here, compressing the sets under consideration canonly improve our objective function. However, we are restricting ourselves to affinely generatingsets and compression may destroy this property (e.g., Example 4). Therefore, our strategy is tokeep compressing as long as affine generation is maintained. To this end we introduce the followingdefinition.Suppose that A ⊇ E , where E = { , e , e , ..., e n } is the standard affine basis of Z n . If A is I -compressed for every I such that C I ( A ) ⊇ E , we say that A is (cid:104)(cid:104) E (cid:105)(cid:105) - compressed . Note that bypart (3) of Lemma 5 every set A containing E , can be turned into an (cid:104)(cid:104) E (cid:105)(cid:105) -compressed set by afinite sequence of such compressions. It turns out that (cid:104)(cid:104) E (cid:105)(cid:105) -compressed sets are very structured. Lemma 7 (Structure of (cid:104)(cid:104) E (cid:105)(cid:105) -compressed sets) . Let A ⊆ Z n be an (cid:104)(cid:104) E (cid:105)(cid:105) -compressed set.(1) A is a shift-minimal downset.(2) A contains a subgroup of maximal size H (cid:47) Z n of the form H = (cid:104) , e , ..., e h (cid:105) .(3) A is { , ..., h, h + i } -compressed for every ≤ i ≤ m = codim H .(4) A ⊆ H + E , i.e. A = H ∪ A ∪ A ∪ ... ∪ A m where A i = A ∩ ( e h + i + H ) .(5) For ≤ i ≤ m , < | A i | < | H | .(6) For ≤ i < j ≤ m , | A i | + | A j | ≤ | H | .(7) If m > , then | A | ≤ (cid:0) m (cid:1) | H | .Proof. The proofs are fairly straightforward.(1) It is a simple observation that both { i } -compressions and { i, j } -compressions preserve E ⊆ A .Hence A must already be compressed with respect to these sets, i.e., a shift-minimal downset.(2) Let h be the maximal dimension of a subgroup contained in A . As shown below in Lemma 8,a subgroup of dimension h must contain an element of Hamming weight at least h . Byshift-minimality e + e + ... + e h ∈ A , and by the downset property H = (cid:104) , e , ..., e h (cid:105) ⊆ A .(3) Denote I = { , ..., h, h + i } . The sets H ∪ { e h + i } and { e h + j } for j (cid:54) = i are initial segments oftheir H I -cosets. These sets cover E and remain included in A through the I -compression.(4) By the downset property, it is sufficient to show e h + i + e h + j / ∈ A for each 1 ≤ i < j ≤ m .Indeed, if A contains e h + i + e h + j then it contains e h + i + H by being { , ..., h, h + j } -compressed.This implies H ∪ ( e h + i + H ) ⊆ A , contrary to the maximality of the subgroup H in A .(5) For the lower bound note that e h + i ∈ E ⊆ A . On the other hand, if | A i | = | H | then H ∪ ( e h + i + H ) ⊆ A , contrary, again, to the maximality of H .(6) Note that e h + j ∈ A , while some lexicographically smaller elements in e h + i + H are notcontained in A . Therefore A can’t be I -compressed for I = { , ..., h, h + i, h + j } . Since it is (cid:104)(cid:104) E (cid:105)(cid:105) -compressed, this means e h + j / ∈ C I ( A ). Equivalently, | A ∩ H I | ≤ | H | , which leads toour claim.(7) If | A i | ≤ | H | for every i , clearly | A | = | H | + (cid:80) mi =1 | A i | ≤ (cid:0) m (cid:1) | H | . Otherwise, | A i | > | H | for some i , thus | A j | ≤ | H | − | A i | < | H | for every j (cid:54) = i . So | A i | + | A j | ≤ | H | for some i and j , and the remaining A j ’s are no bigger than | H | . Lemma 8.
Let H be an h -dimensional subgroup of Z n . Then H contains an element of Hammingweight at least h .Proof. If h = n , take e + e + ... + e n . Otherwise, there exists a basis element e i such that e i / ∈ H .In this case, moving from H to C { i } ( H ) simply deletes e i from the standard basis representationsof H ’s elements, thereby not increasing their Hamming weights. Now note that C { i } ( H ) is an h -dimensional subgroup of (cid:104) , e , ..., e i − , e i +1 , ..., e n (cid:105) , and by induction on n contains an elementof Hamming weight at least h . The Freiman–Ruzsa Theorem in Z n A ⊆ Z n we refer to | (cid:104) A (cid:105) | / | A | as A ’s spanning constant and to K = | A + A | / | A | as its doublingconstant . The Freiman–Ruzsa theorem gives an upper bound on the spanning constant in termsof K . We first review the theorem and some of its quantitative aspects. Then we calculate thebound explicitly, and in particular we determine its correct asymptotics which turns out to beΘ(2 K /K ). We present the proof in two stages, starting with the asymptotic estimates. We findthis presentation convenient, since the proof of the asymptotic bound already contains our mainideas. Freiman’s celebrated theorem [14] states that if A ⊂ Z is a finite subset with | A + A | ≤ K | A | , then A is included in a generalized arithmetic progression, whose size (relative to | A | ) and dimensionare bounded. The bounds depend only on K and not on | A | . Ruzsa [28, 29] has made crucialcontributions to this area. More recently much work was done on similar problems where Z isreplaced by other groups. In particular Ruzsa [31] proved the analogous result for abelian torsiongroups. See [37] for a nice exposition. Theorem 9 (Ruzsa) . Let G be an abelian group in which every element has order at most r . If A is a finite subset of G with | A + A | ≤ K | A | , then A is contained in a coset of a subgroup H (cid:47) G ofsize | H | ≤ f ( r, K ) | A | , where f ( r, K ) ≤ K r K . Better estimates on f ( r, K ) were subsequently found. We denote by F ( r, K ) the smallest boundfor which this statement holds. Note that F ( r, K ) is non-decreasing in K and F ( r,
1) = 1.By considering the case where A is an affine basis of Z K − r we see that F ( r, K ) ≥ r K − O (log K ) (see Example 11 below). This suggests the following conjecture [31]. Conjecture 10 (Ruzsa) . For some C ≥ we have F ( r, K ) ≤ r CK . In an attempt to understand the role of torsion in these phenomena, much work was dedicatedto the special case r = 2, where G = Z n . This work is also motivated by the role that Z n plays indiscrete mathematics and in particular in coding theory [3]. We introduce the following notation: F ( K ) = F (2 , K ) = sup (cid:40) |(cid:104) A (cid:105)|| A | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A ⊆ Z n , n ∈ N , | A + A || A | ≤ K (cid:41) As already observed by Ruzsa [5], for r = 2 his method gives somewhat more, namely F ( K ) ≤ K (cid:98) K (cid:99) − . Later work by Green and Ruzsa [16] gave F ( r, K ) ≤ K r K − , which was againrefined for r = 2 to F ( K ) ≤ O ( K / log K ) by Sanders [32]. Using compressions, Green andTao [17] were able to prove F ( K ) ≤ K + O ( √ K log K ). Note that this confirms Conjecture 10 for r = 2. The best bound so far is due to Konyagin [23] who further improved this method to derive F ( K ) ≤ K + O (log K ) .The range of small K has received some attention as well. In the sub-critical range K <
2, theexact value of F ( K ) is known to be F ( K ) = K for 1 ≤ K < / F ( K ) = K for 7 / ≤ K < K ≤ / F ( K ) ≤ (2 K − / (3 K − K −
1) and for 12 / < K < F ( K ). Example 11 (Independent Points) . Consider the subset: A [ t ] = { , e , e , ..., e t } ⊆ Z t Here, for t ∈ N we have F (cid:18) ( t ) + t +1 t +1 (cid:19) ≥ t t +1 , and by monotonicity one can obtain: F ( K ) ≥ K K (1 − o (1)) .2 Asymptotics of F ( K ) We first prove a new upper bound, which coincides with the construction in Example 11 for t ∈ N . Theorem 12. F (cid:18) ( t ) + t +1 t +1 (cid:19) ≤ t t +1 holds for ≤ t ∈ R . Consequently, F ( K ) ≤ K K (1 − o (1)) . The exponential term 2 K is as in [17, 23], but the polynomial coefficient 1 /K is new. Thusit re-proves Conjecture 10 for r = 2 with C = 2. This bound and Example 11 determine theasymptotics of F ( K ) up to a factor of 2. In the next section we calculate F ( K ) exactly, and showthat the gap is unavoidable and results from the oscillations in F ( K ). Proof.
For an affinely generating subset A ⊂ G = Z n , it is sufficient to prove: | A | = t + 12 t | G | ⇒ | A + A | ≥ (cid:0) t (cid:1) + t + 12 t | G | (1)where 2 ≤ t ∈ R . Since both expressions are monotone in t , the theorem follows.As in [17], the main tool is reduction to compressed sets of some sort. First, since (cid:104) A (cid:105) = G wecan assume that A contains an affine basis for G . But | A | , | A + A | are not affected by invertible affinetransformations, so we may assume without loss of generality E ⊆ A , where E = { , e , e , ..., e n } isthe standard affine basis of G . Now we assume without loss of generality that A is (cid:104)(cid:104) E (cid:105)(cid:105) -compressed.Indeed, supposing (1) holds for (cid:104)(cid:104) E (cid:105)(cid:105) -compressed subsets, we proceed to general subsets inductingon (cid:126) ( A ). Let I ⊆ [ n ] be a set such that E ⊆ C I ( A ) (cid:54) = A . By Lemma 6, | C I ( A ) + C I ( A ) | ≤ | A + A | while | C I ( A ) | = | A | , so A satisfies (1) provided that C I ( A ) does. The inductive argument applies,since (cid:126) ( A ) > (cid:126) ( C I ( A )) by Lemma 5(3).We continue the proof using the structure of (cid:104)(cid:104) E (cid:105)(cid:105) -compressed sets. As in Lemma 7 let H ⊆ A be a maximal subgroup, h = dim H , m = codim H and A i = A ∩ ( e h + i + H ) for 1 ≤ i ≤ m . ByLemma 7(7), | A | ≤ (1 + m/ | H | , and an upper bound on m is given by1 + m m ≥ | A || G | where the case m = 1 follows from the assumption 2 ≤ t .Given m , Lemma 7(4) gives a decomposition of A into m + 1 parts, and we use it to show that A + A is at least ∼ m/ A . This is shown by the following calculation, where allindices go from 1 to m and all unions are disjoint: A = H ∪ (cid:91) i A i ⇒ A + A = H ∪ (cid:91) i ( A i + H ) ∪ (cid:91) i 12 ( | A | − | H | ) ⇒ | A + A | ≥ | H | + m | H | + m − 12 ( | A | − | H | ) = m + 32 · | G | m + m − | A | The right-hand side is decreasing in m in the real interval where (( m + 3) log 2 − / m > | A | / | G | .This interval includes the range of our interest, which is ( m/ / m ≥ | A | / | G | = ( t + 1) / t , orequivalently m ≤ t − 1. Thus, we obtain a lower bound on | A + A | by evaluating this expression at t − 1, namely: | A + A | ≥ ( t − 1) + 32 ( t − | G | + ( t − − · t + 12 t | G | = (cid:0) t (cid:1) + t + 12 t | G | .3 Exact Calculation of F ( K ) Theorem 2, which we will shortly prove, provides an explicit formula of F ( K ). This enables one torederive the asymptotics of F ( K ), and to deduce the following corollary. Corollary 13. Both bounds in the asymptotic inequalities K K (1 − o (1)) ≤ F ( K ) ≤ K K (1 − o (1)) are sharp up to the o (1) terms. It also settles the following conjecture of Diao [6]. Corollary 14. F ( K ) is a piecewise linear function. Figure 1: An illustration of F ( K ) In order to calculate F ( K ), it is useful to consider a related function ˜ K ( ˜ F ), which is defined forrational numbers of the form ˜ F = 2 a /b ≥ K ( ˜ F ) = inf (cid:40) | A + A || A | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A ⊆ Z n , n ∈ N , |(cid:104) A (cid:105)|| A | = ˜ F (cid:41) That is, the minimal doubling constant of an affinely generating set of relative size exactly 1 / ˜ F . Bydefinition F ( K ) = sup { ˜ F | ˜ K ( ˜ F ) ≤ K } . Theorem 12 asserts ˜ K (2 t / ( t + 1)) ≥ (cid:0)(cid:0) t (cid:1) + t + 1 (cid:1) / ( t + 1) or real t ≥ 2, and by Example 11 it is an equality for t ∈ N . In order to analyze ˜ K ( ˜ F ), we refinethe arguments in the proof of Theorem 12, and elaborate on the construction in Example 11. Thisyields a better view of the structure of sets with a small doubling constant. We begin by describingthe extended example. Example 15. For non-negative integers s, t such that s < t , consider the subset: A [ t,s ] = { , e , e , e , ..., e t , e + e , e + e , ..., e + e t − s } ⊆ Z t +12 It is not hard to verify that | A [ t,s ] | = 2( t + 1) − s, | A [ t,s ] + A [ t,s ] | = 2 (cid:18)(cid:18) t (cid:19) + t + 1 (cid:19) − (cid:18) s (cid:19) , (cid:12)(cid:12)(cid:10) A [ t,s ] (cid:11)(cid:12)(cid:12) = 2 t +1 . Therefore: ˜ K (cid:18) t t + 1 − s/ (cid:19) ≤ (cid:0) t (cid:1) + t + 1 − (cid:0) s (cid:1) / t + 1 − s/ K ( ˜ F ) for a discrete sequence of values. When s = 0it reduces to Example 11. However, ˜ K ( ˜ F ) is not necessarily monotone, so we cannot imitate theconclusion of Example 11 and extend the upper bound to general ˜ F . Still, the following argumentdoes the work. Lemma 16 (Sublinearity of ˜ K ( ˜ F )) . If F < F are in ˜ K ’s domain, then ˜ K ( F ) F ≥ ˜ K ( F ) F .Proof. Let F = 2 a /b for some a, b ∈ N . Suppose A ⊆ Z n is an affinely generating set of size | A | = 2 n /F . Let m ∈ N be large enough such that a ≤ n + m < b n + m − a . Consider A (cid:48) = A × Z m , and note that A (cid:48) affinely generates Z n + m and | A (cid:48) | = 2 n + m /F . Since F < F onecan take a subset A ⊆ A (cid:48) of cardinality | A | = b n + m − a = 2 n + m /F . Moreover, by m ’s choice n + m + 1 ≤ | A | , so a subset A which affinely generates Z n + m can be chosen. Now from A + A ⊆ A (cid:48) + A (cid:48) = ( A + A ) × Z m , | A + A || A | · F = | A + A | n = | A (cid:48) + A (cid:48) | n + m ≥ | A + A | n + m = | A + A || A | · F ≥ ˜ K ( F ) F . The task is accomplished by taking the infimum over A . Corollary 17 (Superlinearity of F ( K )) . F ( K ) K ≤ F ( K ) K for every ≤ K < K . Example 15 and Lemma 16 supply an upper bound on ˜ K ( ˜ F ). The following lemma essentiallyclaims that this bound is sharp. Lemma 18 (Formula for ˜ K ( ˜ F )) . Let ˜ F ≥ be of the form a /b where a, b ∈ N , and let s < t bethe unique pair of non-negative integers for which t t + 1 − s/ ≤ ˜ F < t t + 1 − ( s + 1) / Then ˜ K ( ˜ F ) = (cid:0) t (cid:1) + t + 1 − (cid:0) s (cid:1) t · ˜ F Since the function F ( K ) is basically the inverse of ˜ K ( ˜ F ), Theorem 2 is a direct consequenceof Lemma 18. Indeed, Figure 1 is obtained by transposing the graph in Figure 2, and taking themaximum wherever the result is multivalued. We omit further details.One can notice that ˜ K ( ˜ F ) has a more complex structure than F ( K ). Since Theorem 2 employsthe information in Lemma 18 only partially, there may be a quicker way of calculating F ( K ).Nevertheless, we feel that the detailed description of ˜ K ( ˜ F ) is interesting in its own right, and mayshed light on the non-trivial form of F ( K ).The proof of Lemma 18 pursues the analysis in Theorem 12’s proof, involving more reductionsteps which preserve | A | without increasing | A + A | . Through these reductions the structure of A K ( ˜ F ) becomes similar to Example 15, so that its doubling constant can be calculated explicitly. We startwith two reductions which can be formulated separately in terms of integer partitions. All of thefollowing will be motivated and applied later, in the proof of the lemma.A non-increasing sequence of positive integers a ...a m is an integer partition of a = (cid:80) i a i into m parts, and for short an m - partition of a . Recall the Hopf–Stiefel function a ◦ b from Section 2.We are interested in the minimum of (cid:80) ≤ i A compressed m -partition of a that minimizes (cid:80) i 2. For every two positive integers m ≤ a , there exists a unique quasi-fair m -partition of a .3. If a ...a m and a (cid:48) ...a (cid:48) m are the quasi-fair m -partitions of a ≤ a (cid:48) , then a i ≤ a (cid:48) i for all i .4. A sub-partition (in the sense of a sub-sequence) of a quasi-fair partition is quasi-fair.Now we are ready to state the second reduction. Lemma 20. The minimum of (cid:80) i 2, and consequently: m (cid:88) i =1 ( m − i ) · a i = m (cid:88) i =1 ( m − i ) · a i + ( m − (cid:32) m (cid:88) i =1 a (cid:48) i − m (cid:88) i =1 a i (cid:33) = m (cid:88) i =1 ( m − i ) · a (cid:48) i − m (cid:88) i =2 ( i − a i − a (cid:48) i ) < m (cid:88) i =1 ( m − i ) · a (cid:48) i With these reductions in hand, we can complete the calculation of ˜ K ( ˜ F ). roof. (of Lemma 18) Let (cid:104) A (cid:105) = G = Z n . Lemma 18 is proved by showing the following lowerbound on | A + A | , which is reached by Example 15 and Lemma 16: t + 1 − ( s + 1) / t < | A || G | ≤ t + 1 − s/ t ⇒ | A + A || G | ≥ (cid:0) t (cid:1) + t + 1 − (cid:0) s (cid:1) t (2)If | A | > | G | , then by the pigeonhole principle A + A = G , as required in the cases t = 1 , 2. Hencewe may assume | A | ≤ | G | and t ≥ A is (cid:104)(cid:104) E (cid:105)(cid:105) -compressed, and therefore by Lemma 7 has the following properties: • There exists a subgroup H = (cid:104) , e , ..., e h (cid:105) such that A = H ∪ A ∪ A ∪ ... ∪ A m , where A i = A ∩ ( e h + i + H ) and m = codim H . • / m < | A | / | G | ≤ (1 + m ) / m . By the assumptions t +22 t +1 < | A | / | G | ≤ , we can write1 < m < t . • Each A i is a lexicographic initial segment of e i + H . Therefore A is uniquely determined bythe sequence a , ..., a m where a i = | A i | . Note that 0 < a i < h . • By shift-minimality a ≥ a ≥ ... ≥ a m . In other words, a , ..., a m is a partition of a = | A | − | G | / m .As in Theorem 12, we use these properties to write A + A as a disjoint union of its intersectionswith H -cosets, which are of three forms: H , H + A i and A i + A j . Since the A i ’s are initial segmentsof their cosets, the sumsets of the third form can be expressed via the Hopf–Stiefel function: | A + A | = | H | + m | H | + (cid:88) ≤ i Note that in the cases j = m − j = m we can choose j (cid:48) = 0 and k (cid:48) = k + 1 as well.We could avoid this freedom of choice by not permitting j = 0, but since it does not affect theresulting | A + A | , we allow both ways.All that remains now is to show that, as in Theorem 12, to minimize | A + A | we should make m as large as possible, i.e., m = t − 1. The proof is by induction on t − m : • Suppose m = t − 1. We check that (2) holds.2 m − ( s + 1)2 · | G | m +1 < | A | − | G | m ≤ m − s · | G | m +1 Denote k = dim G − m − j = m − s , and observe that 0 ≤ j ≤ m . Then the aboveexpression for the minimal | A + A | becomes | A + A | = m + 12 m · | G | + (cid:20)(cid:18) m (cid:19) − (cid:18) m − ( m − s )2 (cid:19)(cid:21) · | G | m +1 = (cid:0) t (cid:1) + t + 1 − (cid:0) s (cid:1) t · | G | Suppose m < t − 1. The above discussion yields a compressed set A , such that a ...a m is thequasi-fair quasi-dyadic partition of | A | − | H | , and | A + A | is minimal given m , and equals: | A + A | = m + 12 m | G | + (cid:88) ≤ i An examination of the proof reveals two kinds of reduction steps. Either A is compressedwithout changing (cid:104) A (cid:105) , or we find a set A (cid:48) where | A (cid:48) | = | A | and | A (cid:48) + A (cid:48) | is substantially smallerthan | A + A | . Hence, the proof actually provides a characterization of the extremal case, up tocompressions that preserve (cid:104) A (cid:105) and | A + A | . What is the smallest possible cardinality of A + B if A, B ⊆ G = Z n are two affinely spanningsubsets of given cardinalities? In this section we prove Theorem 1, which gives an essentiallycomplete answer. In addition we establish a new isoperimetric inequality, which is used in theproof. But first, we make some remarks concerning the theorem. Remarks (on Theorem 1) . 1. Tightness: Consider ( | A | / | G | , | B | / | G | , | A + B | / | G | ) as a point in [0 , . The Hamming ballsconstruction shows that the bound goes through the points of the form: (cid:32) t t , t + ... + (cid:0) tk (cid:1) t , t + ... + (cid:0) tk +1 (cid:1) t (cid:33) ≤ k < t An inspection of Figure 3 shows that all points properly inside their convex hull are strictlybelow the bound, and hence cannot be realized by such sets. In other words, further improve-ments of the bound will be local in nature.2. The formulation of the theorem apparently breaks the symmetry and doesn’t require (cid:104) B (cid:105) = G .Still, there is an asymmetry in the result as well, and the theorem is of interest mostly when | A | ≤ | B | . See also the remark after the proof.3. In order to simplify the statement of the theorem, t is defined as the largest positive integersuch that | A | / | G | ≤ ( t + 1) / t . However, the only assumption on t which the proof actuallyuses is: t + 22 t +1 < | A || G | The theorem can, therefore, be applied as well with t larger than in the given formulation.As Figure 3 shows, the resulting bound would be weaker, but may still be useful in certaincontexts. Theorem 1 implies that a large enough number of large enough affinely generating sets mustadd up to the whole group: Corollary 21. Suppose that (cid:104) A (cid:105) = (cid:104) A (cid:105) = ... = (cid:104) A m (cid:105) = G = Z n with | A i | / | G | > ( m + 2) / m +1 for all i . Then A + A + ... + A m = G .Proof. We repeatedly apply the theorem with A = A i and B = A + ... + A i − for all 1 ≤ i ≤ m to conclude | A + A + ... + A i || G | > (cid:0) m +10 (cid:1) + (cid:0) m +11 (cid:1) + ... + (cid:0) m +1 i (cid:1) m +1 Indeed, in view of remark 3 above and the assumption on the cardinalities, we may choose t = m +1,and then k = i − w > | A + A + ... + A m − | + | A m | > | G | the proof is completed by the pigeonhole principle, | A | + | B | > | G | ⇒ A + B = G .The special case of Corollary 21 where all A i are identical is due to Lev [25], following aconjecture of Zemor [40]. Taking A i = D m +11 × Z n − m +12 for each i shows that the assumption onthe cardinalities is sharp. We are inspired by Frankl’s short inductive proof [13] of Harper’s theorem [18]. heorem 22 (Harper’s Inequality) . Suppose A ⊆ Z n .If for ≤ k ≤ n integer and ≤ p ≤ real | A | = (cid:18) nn (cid:19) + (cid:18) nn − (cid:19) + ... + (cid:18) nk + 1 (cid:19) + p (cid:18) nk (cid:19) then | A + D n | ≥ (cid:18) nn (cid:19) + (cid:18) nn − (cid:19) + ... + (cid:18) nk (cid:19) + p (cid:18) nk − (cid:19) In simple terms this theorem says that Hamming balls solve the vertex-isoperimetric problem inthe hypercube. However, it also deals, to varying degrees depending on the version of the theorem,with sets of cardinalities strictly between | D nk − | and | D nk | . A stronger version would replace the lastsummand of each expression with (cid:0) xk (cid:1) and (cid:0) xk − (cid:1) respectively, where x ∈ [ k, n ] is real. The optimalformulation due to Katona [22] and Kruskal [24] is stated in terms of the k -cascade representations (cid:0) a k k (cid:1) + (cid:0) a k − k − (cid:1) + ... and (cid:0) a k k − (cid:1) + (cid:0) a k − k − (cid:1) + ... respectively. Frankl’s method yields all three formulations.Frankl’s proof employs several useful operators on set-systems. As usual, we freely move betweenthe set-theoretic terminology of 2 [ n ] and the algebraic language of Z n . The push-down operator T i and the shift operator S ij have already appeared in Section 2. The upper and the lower shadow operators act on a set-system F ⊆ [ n ] by δ F = { J ∪ { i } | J ∈ F , i / ∈ J } ∂ F = { J \ { i } | J ∈ F , i ∈ J } respectively. For downsets, the notion of the shadow is close to that of the neighborhood in thetheorem. If C ⊆ Z n is a non-empty downset, then C + D n = δC ∪ { } . Note that always 0 / ∈ δA .Another useful operation on set-systems is classification by n , denoted by: F − = { J | J ∈ F , n / ∈ J }F + = { J \ { n } | J ∈ F , n ∈ J } When A ⊆ Z n , we regard A + and A − as subsets of Z n − .Following Frankl [13], we proceed with two lemmas regarding properties of shifts and shadows. Lemma 23. Suppose C ⊆ Z n is a shift-minimal downset.(1) δ ( C + ) ⊆ ( δC ) + = C − with equality iff C = ∅ (2) δ ( C − ) = ( δC ) − Proof. Examine the effect of the operators on the representation of some x ∈ C with the standardbasis e , ..., e n .In both δ ( C + ) and ( δC ) + , some e i is added and e n is removed. However, in δ ( C + ) certainly i (cid:54) = n since C + lives in Z n − , while in ( δC ) + it is possible that i = n . Hence δ ( C + ) ⊆ ( δC ) + .By shift-minimality C is closed under these swaps, thus ( δC ) + ⊆ C − . Moreover, every element of C − is obtained by adding e n and then deleting it, so there is equality. However, δ ( C + ) is strictlysmaller since 0 ∈ C − \ δ ( C + ) unless C is empty.For δ ( C − ) = ( δC ) − , note that both sets consist of elements of the form x + e i for x ∈ C and i < n , where e i and e n do not appear in x ’s standard representation.The following lemma is well known. See e.g. [11, 21]. Here we prove it as a special case of thecompression machinery. Lemma 24. For all A ⊆ Z n and ≤ i, j ≤ n such that i (cid:54) = j ,(1) δ ( S ij A ) ⊆ S ij ( δA ) (2) ∂ ( S ij A ) ⊆ S ij ( ∂A ) roof. By passing from A to (cid:80) i e i − A , it is enough to prove only one of the inclusions. Denote A = (cid:83) nk =0 A k where A k = A ∩ (cid:0) D nk \ D nk − (cid:1) . Note that we can work with each A k separately. Onecan write δ ( S ij A k ) = (cid:0) D n + C ij (cid:0) A k ∪ D nk − (cid:1)(cid:1) \ D nk , and S ij ( δA k ) = C ij (cid:0) D n + (cid:0) A k ∪ D nk − (cid:1)(cid:1) \ D nk , yielding our claim by Lemma 6, since D n + C ij ( B ) = C ij ( D n ) + C ij ( B ) ⊆ C ij ( D n + B ).Our isoperimetric inequality concerns a family of non-empty downsets C ...C l ⊆ Z n , rather thana single one. For the volume and the shadow we take the average quantities, denoted by:E [ C ] = 1 l l (cid:88) m =1 | C m | E [ δC ] = 1 l l (cid:88) m =1 | δ ( C m ) | It is hard to make a meaningful statement about these average quantities without limiting thedownsets somehow. To see this, consider what happens when each C m is either full or empty.We limit the variability of the downsets by assuming the antichain condition . Namely, we requirethat for each i and j , C i \ C j is an antichain with respect to set-systems inclusion, or equivalently C j ⊇ ∂C i . Proposition 25. Suppose C ...C l ⊆ Z n is a family of downsets which satisfies the antichaincondition. If E[ C ] = (cid:18) n (cid:19) + (cid:18) n (cid:19) + ... + (cid:18) nk − (cid:19) + p (cid:18) nk (cid:19) for some integer k ≥ and real number ≤ p < , then E [ δC ] ≥ (cid:18) n (cid:19) + (cid:18) n (cid:19) + ... + (cid:18) nk (cid:19) + p (cid:18) nk + 1 (cid:19) Since for non-empty downsets C + D n = { } ∪ δC , the corresponding inequality in the languageof neighborhoods is as follows. Corollary 26. In the setting of Proposition 25, if C ...C l are non-empty then E [ C + D n ] ≥ (cid:18) n (cid:19) + (cid:18) n (cid:19) + ... + (cid:18) nk (cid:19) + p (cid:18) nk + 1 (cid:19) Proof. (of Proposition 25) We may assume that the downsets are shift-minimal. Indeed, for eachdownset C m clearly S ij C m is a downset of the same size, while | δ ( S ij C m ) | ≤ | S ij ( δC m ) | = | δC m | byLemma 24. If C m (cid:48) \ C m is an antichain, then C m ⊇ ∂C m (cid:48) , hence S ij C m ⊇ S ij ( ∂C m (cid:48) ) ⊇ ∂ ( S ij C m (cid:48) )by Lemma 24 again, and hence S ij C m (cid:48) \ S ij C m is an antichain as well. In conclusion, S ij C ...S ij C l satisfy the antichain condition, E[ C ] = E[ S ij C ] and E[ δC ] ≥ E[ δ ( S ij C )]. After a finite sequence ofshifts the downsets are all shift-minimal, since for a proper shift (cid:80) m (cid:126) ( S ij C m ) < (cid:80) m (cid:126) ( C m ).The case k = 0 is established separately. Note that in this case E[ C ] < 1, hence C m = ∅ forsome m . Actually, this is a sufficient condition for k = 0, because all other downsets are either ∅ or { } by the antichain condition. Since δ { } = { e , ..., e n } , clearly E [ δC ] = n · E[ C ] as required.Following Frankl, we proceed by induction on n . By convention (cid:0) nk (cid:1) = 0 for n < k . Thus, for n = 0 the lemma is vacuously satisfied by E [ δC ] ≥ k and n , we employ the induction hypothesis on the families C − ...C − l and C +1 ...C + l in Z n − . It is easily checked that given a downset C m , the sets C + m and C − m are downsets as well.In addition, if C m (cid:48) \ C m is an antichain, then so are its two parts, C − m (cid:48) \ C − m and C + m (cid:48) \ C + m , hencethe new families satisfy the antichain condition.By the induction hypothesis on C +1 ...C + l ⊆ Z n − , at least one of the following must hold:E (cid:2) C + (cid:3) < (cid:18) n − (cid:19) + (cid:18) n − (cid:19) + ... + (cid:18) n − k − (cid:19) + p (cid:18) n − k − (cid:19) E (cid:2) δ (cid:0) C + (cid:1)(cid:3) ≥ (cid:18) n − (cid:19) + (cid:18) n − (cid:19) + ... + (cid:18) n − k − (cid:19) + p (cid:18) n − k (cid:19) se E [ C − ] = E[ C ] − E [ C + ] and Pascal’s rule in the first case, or E [ C − ] ≥ δ ( C + )] byLemma 23(1) in the second one, to deduce:E (cid:2) C − (cid:3) ≥ (cid:18) n − (cid:19) + (cid:18) n − (cid:19) + ... + (cid:18) n − k − (cid:19) + p (cid:18) n − k (cid:19) Note that since k > C m is non-empty, so there is proper inclusion in the lemma, which yieldsthe extra 1 in the calculation. By the induction hypothesis on C − ...C − l ⊆ Z n − :E (cid:2) δ (cid:0) C − (cid:1)(cid:3) ≥ (cid:18) n − (cid:19) + (cid:18) n − (cid:19) + ... + (cid:18) n − k (cid:19) + p (cid:18) n − k + 1 (cid:19) By Lemma 23, E (cid:2) δC (cid:3) = E (cid:2) ( δC ) − (cid:3) + E (cid:2) ( δC ) + (cid:3) = E (cid:2) δ ( C − ) (cid:3) + E (cid:2) C − (cid:3) , hence by Pascal’s rule:E [ δC ] ≥ (cid:18) n (cid:19) + (cid:18) n (cid:19) + ... + (cid:18) nk (cid:19) + p (cid:18) nk + 1 (cid:19) Proof. (of Theorem 1) The general idea is similar to the case A + A discussed in the previoussection. By applying various compressions, the sets A and B acquire certain structural properties.These, in turn, allow us to derive estimates on the cardinality of A + B .Lemma 6 asserts that compressions do not increase sumsets: | C I ( A ) + C I ( B ) | ≤ | A + B | holdswhile | C I ( A ) | = | A | and | C I ( B ) | = | B | . Thus, in the search for a lower bound for | A + B | , onecan first apply a compression C I on A and B simultaneously. Since (cid:104) A (cid:105) = G , we may suppose E = { , e , e , ..., e n } ⊆ A and restrict ourselves only to compressions that preserve the inclusion E ⊆ A . By Lemma 5(3), if a compression C I changes either A or B , then (cid:126) ( A ) + (cid:126) ( B ) strictlydecreases. It follows that every sequence of such compressions must terminate. In conclusion, wecan assume that both A and B are invariant under these compressions, or for short (cid:104)(cid:104) E ⊆ A (cid:105)(cid:105) - compressed . This implies that B is I -compressed for every I ⊆ [ n ] such that A is I -compressed.Lemma 7 provides a description of A under this assumption. In particular, H ⊆ A ⊆ H + E forsome subgroup H = (cid:104) , e , ..., e h (cid:105) . We next derive some structural properties of B . Lemma 27. Suppose A, B ⊆ G = Z n are (cid:104)(cid:104) E ⊆ A (cid:105)(cid:105) -compressed. Let H ⊆ A be as in Lemma 7.Consider G/H ∼ = Z m where m = n − h = codim H , with the basis { e h +1 + H, ..., e h + m + H } andthe partial order of the corresponding set-system. For ≤ j ≤ | H | let C j = (cid:110) H (cid:48) ∈ G/H (cid:12)(cid:12)(cid:12) | B ∩ H (cid:48) | ≥ j (cid:111) Then C ...C | H | are downsets, and satisfy the antichain condition.Proof. By Lemma 7(3), A is { , ..., h, h + i } -compressed for 1 ≤ i ≤ m , and therefore so is B .Let H (cid:48) ≺ H (cid:48)(cid:48) be adjacent H -cosets in the partial order. H (cid:48)(cid:48) = e h + i + H (cid:48) for some 1 ≤ i ≤ m .Since B is { , ..., h, h + i } -compressed, B ∩ ( H (cid:48) ∪ H (cid:48)(cid:48) ) must be an initial segment of H (cid:48) ∪ H (cid:48)(cid:48) . Notethat all H (cid:48) elements are lexicographically smaller than those of H (cid:48)(cid:48) . Consequently, if B ∩ H (cid:48)(cid:48) (cid:54) = ∅ then necessarily H (cid:48) ⊆ B . In other words, H (cid:48)(cid:48) ∈ C ⇒ H (cid:48) ∈ C | H | for each such pair.In particular, C j is a downset because H (cid:48)(cid:48) ∈ C j ⊆ C ⇒ H (cid:48) ∈ C | H | ⊆ C j , and C j \ C k is anantichain since C j \ C k ⊆ C \ C | H | (cid:54)⊇ { H (cid:48) , H (cid:48)(cid:48) } .We can conclude now the proof of Theorem 1 in the following three steps:1. We use the structure of the compressed sets to find new expressions for the cardinalities of B and A + B . Let C ...C | H | be as in Lemma 27. By interchanging the order of summation: | B | = (cid:88) H (cid:48) ∈ G/H | B ∩ H (cid:48) | = (cid:88) H (cid:48) ∈ G/H (cid:110) j ∈ N (cid:12)(cid:12)(cid:12) | B ∩ H (cid:48) | ≥ j (cid:111) = | H | (cid:88) j =1 | C j | We estimate | A + B | in a similar fashion. For 1 ≤ j ≤ | H | , suppose H (cid:48)(cid:48) ∈ δ ( C j ) ∪ { H } . Weshow that A + B intersects H (cid:48)(cid:48) in j elements at the least: If H (cid:48)(cid:48) = H , use H ⊆ A and 0 ∈ B (cid:54) = ∅ to obtain | ( A + B ) ∩ H (cid:48)(cid:48) | ≥ | ( H + 0) ∩ H | ≥ j . • Otherwise H (cid:48)(cid:48) = e h + i + H (cid:48) for some H (cid:48) ∈ C j and 1 ≤ i ≤ m = codim H . Since e h + i ∈ E ⊆ A , clearly | ( A + B ) ∩ H (cid:48)(cid:48) | ≥ | ( e h + i + B ) ∩ ( e h + i + H (cid:48) ) | = | B ∩ H (cid:48) | ≥ j .Consequently: | A + B | = (cid:88) H (cid:48)(cid:48) ∈ G/H (cid:110) j ∈ N (cid:12)(cid:12)(cid:12) | ( A + B ) ∩ H (cid:48)(cid:48) | ≥ j (cid:111) ≥ | H | (cid:88) j =1 | δ ( C j ) ∪ { H }| 2. We use the isoperimetric inequality in order to obtain a lower bound on | A + B | given m and | B | . Let 0 ≤ k ≤ m and w ∈ [ − , 1] be such that: | B | = (cid:0) m +10 (cid:1) + (cid:0) m +11 (cid:1) + ... + (cid:0) m +1 k (cid:1) + w (cid:0) mk (cid:1) m +1 · | G | We substitute | B | = (cid:80) | C j | in the left-hand side, apply Pascal’s rule to (cid:0) m +11 (cid:1) ... (cid:0) m +1 k (cid:1) on theright-hand side, and divide both by | H | = | G | / m , to obtain:E[ C ] = 1 | H | | H | (cid:88) j =1 | C j | = | B || H | = (cid:18) m (cid:19) + (cid:18) m (cid:19) + ... + (cid:18) mk − (cid:19) + 1 + w (cid:18) mk (cid:19) Now by Proposition 25E [ { H } ∪ δC ] ≥ (cid:18) m (cid:19) + (cid:18) m (cid:19) + ... + (cid:18) mk (cid:19) + 1 + w (cid:18) mk + 1 (cid:19) where the union is disjoint since always H / ∈ δC j . In terms of A and B , this implies: | A + B | ≥ (cid:0) m +10 (cid:1) + (cid:0) m +11 (cid:1) + ... + (cid:0) m +1 k (cid:1) + (cid:0) m +1 k +1 (cid:1) + w (cid:0) mk +1 (cid:1) m +1 · | G | 3. What values can m = codim H take? By Lemma 7(7), (cid:0) m (cid:1) / m ≥ | A | / | G | , where thecase m = 1 is separately deduced from the assumption | A | / | G | ≤ / 4. On the other hand,by the theorem’s assumption on t , | A | / | G | > ( t + 2) / t +1 = (cid:0) t (cid:1) / t . Since the sequence (cid:0) n (cid:1) / n is monotone, we infer m < t .The theorem is obtained by plugging m = t − m the bound is even higher, as demonstrated in Figure 3. Indeed, for each1 ≤ i ≤ t − 2, the graph of the lower bound on | A + B | given m = i is concave down bythe log-concavity of the binomial coefficients, (cid:0) mk (cid:1) / (cid:0) mk − (cid:1) ≥ (cid:0) mk +1 (cid:1) / (cid:0) mk (cid:1) . Thus the graph of m = i + 1, which connects the midpoints of adjacent segments in the m = i graph, must belower. Remark. By the Hamming balls construction, the lower bound we have found is optimal on abiparametric discrete family of points. In view of our treatment of F ( K ) in the previous section,we expect that at intermediate points better bounds should be provable.There are three points where our approach to Theorem 1 may be suboptimal: the isoperimetricinequality we use is not always perfectly tight, the addition of H ∪ E instead of the whole of A ,and dropping the assumption on B ’s affine span.It is perhaps worth remarking that the machinery of compressions can still be applied underthe assumption (cid:104) A (cid:105) = (cid:104) B (cid:105) = G . This is done by showing that without loss of generality we mayassume that A and B are simultaneously compressed such that they include a common affine basis.Here is a brief outline of how this is done. First, partition A and B into their intersections withcosets of (cid:104) ( A − A ) ∩ ( B − B ) (cid:105) . These parts can be translated without increasing | A + B | , such that A − A and B − B include a common basis of G . Then apply { i } -compressions with respect to thisbasis, until it is included in A ∩ B . Acknowledgment I would like to thank my advisor, Professor Nati Linial, for his patient and helpful guidance duringthe research and the preparation of this manuscript. References [1] B Bollob´as and I Leader. Sums in the grid. Discrete Mathematics , 162(1-3):31–48, 1996.[2] A L Cauchy. Recherches sur les nombres. J. ´Ecole Polytechnique , 9:99–123, 1813.[3] G Cohen and G Z´emor. Subset sums and coding theory, structure theory of set addition. Ast´erisque , 258, 1999.[4] H Davenport. On the addition of residue classes. Journal of the London Mathematical Society , 1(1):30, 1935.[5] J M Deshouillers, F Hennecart, and A Plagne. On small sumsets in ( Z / Z ) n . Combinatorica , 24(1):53–68,2004.[6] H Diao. Freiman–Ruzsa-type theory for small doubling constant. Mathematical Proceedings of the CambridgePhilosophical Society , 146(2):269–276, 2009.[7] S Eliahou and M Kervaire. Sumsets in vector spaces over finite fields. Journal of Number Theory , 71(1):12–39,1998.[8] S Eliahou and M Kervaire. Minimal sumsets in infinite abelian groups. Journal of Algebra , 287(2):449–457,2005.[9] S Eliahou and M Kervaire. Old and new formulas for the Hopf–Stiefel and related functions. ExpositionesMathematicae , 23(2):127–145, 2005.[10] S Eliahou, M Kervaire, and A Plagne. Optimally small sumsets in finite abelian groups. Journal of NumberTheory , 101(2):338–348, 2003.[11] P Frankl. A new short proof for the Kruskal–Katona theorem. Discrete Mathematics , 48(2-3):327–329, 1984.[12] P Frankl. The shifting technique in extremal set theory. Surveys in combinatorics , 123:81–110, 1987.[13] P Frankl. A lower bound on the size of a complex generated by an antichain. Discrete mathematics , 76(1):51–56,1989.[14] G A Freiman. Foundations of a structural theory of set addition (translated from the Russian). Translationsof Mathematical Monographs , 37, 1973.[15] R J Gardner and P Gronchi. A Brunn–Minkowski inequality for the integer lattice. Transactions – AmericanMathematical Society , 353(10):3995–4024, 2001.[16] B Green and I Z Ruzsa. Sets with small sumset and rectification. Bulletin of the London Mathematical Society ,38(1):43, 2006.[17] B Green and T Tao. Freiman’s theorem in finite fields via extremal set theory. Combinatorics, Probability andComputing , 18(03):335–355, 2009.[18] L H Harper. Optimal numberings and isoperimetric problems on graphs. Journal of Combinatorial Theory ,1(3):385–393, 1966.[19] F Hennecart and A Plagne. On the subgroup generated by a small doubling binary set. European Journal ofCombinatorics , 24(1):5–14, 2003.[20] H Hopf. Ein toplogischer Beitrag zur reellen Algebra. Commentarii Mathematici Helvetici , 13(1):219–239, 1940.[21] G O H Katona. Intersection theorems for systems of finite sets. Acta Mathematica Hungarica , 15(3):329–337,1964.[22] G O H Katona. The hamming-sphere has minimum boundary. Studia Sci. Math. Hungar , 10(1-2):131–140,1975.[23] S V Konyagin. On the Freiman theorem in finite fields. Mathematical Notes , 84(3):435–438, 2008.[24] J B Kruskal. The number of simplices in a complex. Mathematical optimization techniques , page 251, 1963.[25] V F Lev. Generating binary spaces. Journal of Combinatorial Theory, Series A , 102(1):94–109, 2003.[26] V F Lev. Critical pairs in abelian groups and Kemperman’s structure theorem. Int. J. Number Theory ,2(3):379–396, 2006.[27] A Pfjster. Zur Darstellung von–1 als Summe von Quadraten in einem K¨orper. Journal of the London Mathe-matical Society , 1(1):159, 1965.[28] I Z Ruzsa. Arithmetical progressions and the number of sums. Periodica Mathematica Hungarica , 25(1):105–111,1992.[29] I Z Ruzsa. Generalized arithmetical progressions and sumsets. Acta Mathematica Hungarica , 65(4):379–388,1994.[30] I Z Ruzsa. Sum of sets in several dimensions. Combinatorica , 14(4):485–490, 1994.[31] I Z Ruzsa. An analog of Freiman’s theorem in groups. Ast´erisque , pages 323–326, 1999. 32] T Sanders. A note on Freiman’s theorem in vector spaces. Combinatorics, Probability and Computing ,17(02):297–305, 2008.[33] R Schneider. Convex bodies: the Brunn–Minkowski theory , volume 44. Cambridge Univ Pr, 1993.[34] D B Shapiro. Compositions of quadratic forms , volume 33. Walter De Gruyter Inc, 2000.[35] E Stiefel. ¨Uber Richtungsfelder in den projektiven R¨aumen und einen Satz aus der reellen Algebra. CommentariiMathematici Helvetici , 13(1):201–218, 1940.[36] T Tao and V Vu. Additive combinatorics. Cambridge Studies in Advanced Mathematics , 105, 2006.[37] E Viola. Selected results in additive combinatorics: An exposition. Electronic Colloquium on ComputationalComplexity (ECCC) , 14(103), 2007.[38] A G Vosper. The critical pairs of subsets of a group of prime order. Journal of the London MathematicalSociety , 1(2):200, 1956.[39] S Yuzvinsky. Orthogonal pairings of euclidean spaces. The Michigan Mathematical Journal , 28(2):131–145,1981.[40] G Z´emor. An extremal problem related to the covering radius of binary codes. Algebraic Coding , pages 42–51,1992.[41] G Z´emor. Subset sums in binary spaces. European journal of combinatorics , 13(3):221–230, 1992., 13(3):221–230, 1992.