aa r X i v : . [ m a t h . C O ] A p r A STRUCTURAL APPROACH TO SUBSET-SUM PROBLEMS
VAN VU
Abstract.
We discuss a structural approach to subset-sum problems in ad-ditive combinatorics. The core of this approach are Freiman-type structuraltheorems, many of which will be presented through the paper. These resultshave applications in various areas, such as number theory, combinatorics andmathematical physics. Introduction
Let A = { a , a , . . . } be a subset of an additive group G (all groups discussed inthis paper will be abelian). Let S A be the collection of subset sums of AS A := { X x ∈ B x | B ⊂ A, | B | < ∞} . Two related notions that are frequently considered are lA := { a + · · · + a l | a i ∈ A } l ∗ A := { a + · · · + a l | a i ∈ A, i = j } . We have the trivial relations l ∗ A ⊂ lA and ∪ l l ∗ A = S A . One can have similar definitions for A being a sequence (repetitions allowed). Example. A = { , , } , G = Z , 2 A = { , , , , , } , 2 ∗ A = { , , } , S A = { , , , } . A = { , , } , G = Z , 2 A = G , 2 ∗ A = { , , } = S A . V. Vu is supported by NSF Career Grant 0635606.
Now let A be a sequence: A = { , , } , G = Z , 3 A = { , , , } , 3 ∗ A = { } , S A = { , , , , , } .Notice that for a large l , lA can be significantly different from S A and l ∗ A . Ingeneral, it is easier to handle than the later two.Many basic problems in additive combinatorics have the following form: If A is sufficiently dense in G , then S A (or l ∗ A or lA ) contains a special element(such as or a square), or a large structure (such as a long arithmetic progression G itself ). The main question is to find the threshold for “dense”. As examples, we presentbelow a few well-known results/problems in the area. In the whole paper, we aregoing to focus mostly on two special cases: (1) G = Z p , where Z p denotes the cyclicgroup of residues modulo a large prime p ; (2) G = Z , the set of integers.Following the literature, we say that A is zero-sum-free if 0 / ∈ S A . Furthermore, A is complete if S A = G and incomplete otherwise. The asymptotic notation is usedunder the assumption that | A | → ∞ .A basic result concerning zero-sum-free sets is the following theorem of Olson [53]and Szemer´edi [60] from the late 1960s, addressing a problem of Erd˝os and Heil-bronn [23]. Theorem 1.1. (Olson-Szemer´edi) Let A be a subset of Z p with cardinality C √ p ,for a sufficiently large constant C . Then S A contains zero. To see that order √ p is necessary, consider A := { , , . . . , n } , where n ≈ √ p isthe largest integer such that 1 + · · · + n < p .Concerning completeness, Olson [52], proved the following result Theorem 1.2. (Olson) Any subset A of Z p with cardinality at least √ p − is complete. To see that the bound is close to optimal, take A := {− m, . . . , − , , , . . . , m } where m is the largest integer such that 1 + · · · + m < ⌊ p/ ⌋ .Another classical result concerning zero sums is that of Erd˝os-Ginburg-Ziv [42],again from the 1960s. Theorem 1.3. (Erd˝os-Ginburg-Ziv) If A is a sequence of p − elements in Z p ,then p ∗ A contains zero. This theorem is sharp by the following example: A = { [ p − , [ p − } Furthermore,instead of 0 and 1, one can use any two different elements of Z p . (Here and later x [ k ] means x appears with multiplicity k in A .) STRUCTURAL APPROACH TO SUBSET-SUM PROBLEMS 3
Now we discuss two problems involving integers. Set [ n ] := { , , . . . , n } . An oldand popular conjecture concerning subset sums of integers is Folkman’s conjecture,made in 1966 [25]. Folkman’s conjecture is a strengthening of a conjecture by Erd˝os[20] about finding a necessary and sufficient condition for a sequence A such that S A contains all but finite exception of the positive integers. Conjecture 1.4. (Folkman’s conjecture) The following holds for any sufficientlylarge constant C . Let A be an strictly increasing sequence of positive integers with(asymptotic) density at least C √ n (namely, | A ∩ [ n ] | ≥ C √ n for all sufficientlylarge n ). Then S A contains an infinite arithmetic progression. Cassels [10] and Erd˝os [20] showed that density √ n is indeed needed; thus, Folk-man’s conjecture is sharp up to the value of C . For more discussion about Folk-man’s conjecture and its relation with Erd˝os’ conjecture, we refer to [25] and themonograph [21] by Erd˝os and Graham.Finally, a problem involving a non-linear relation, posed by Erd˝os in 1986 [19]. Problem 1.5. (Erd˝os’ square-sum-free problem) A set A of integers is square-sum-free if S A does not contain a square. Find the largest size of a square-sum-freesubset of [ n ] . Erd˝os observed that one can construct such a square-sum-free subset of [ n ] withat least Ω( n / ) elements. To see this, consider A := { q, q, . . . , kq } with q prime,( k + 1) k < q , kq ≤ n . Since the sum of all elements of A is less than q , S A doesnot contain a square. Erd˝os [19] conjectured that the truth is close to this lowerbound.Problems involving subset sums such as the above (and many others) have beenattacked, with considerable success, using various techniques: combinatorial, har-monic analysis, algebraic etc. The reader who is interested in these techniques maywant to look at [3, 57, 64, 48] and the references therein.The goal of this paper is to introduce the so-called “structural approach”, whichhas been developed systematically in recent years. This approach is based on thefollowing simple plan Step I: Force a structure on A . In this step, one tries to show the following: If A isrelatively dense (close to the conjectured threshold but not yet there) and S A doesnot contain the desired object, then A has a very special structure.Alternately, one can can try to Step I’: Find a structure in S A . If A is relatively dense (again close to the conjec-tured threshold but not yet there) then S A contains a special structure. Step II: Completion.
Since | A | is still below the threshold, we can add (usuallymany) new elements to A . Using these elements together with the existing struc-ture, one can, in most cases, obtain the desired object in a relatively simple manner. VAN VU
The success of the method depends on the quality of the information we can obtainon the structure of A (or S A ) in Step I (or I’). In several recent studies, it has turnedout that one can frequently obtain something close to a complete characterization of these sets. Thanks to these results, one is able to make considerable progresseson many old problems and also reprove and strengthen several existing ones (with abetter understanding and a complete classification of the extremal constructions).The rest of this paper is devoted to the presentation of these structural theoremsand their representative applications.2. Freiman’s structural theorem
A corner stone in additive combinatorics is the structural theorem of Freiman (some-time referred to as Freiman’s inverse theorem), which writes down the structure ofsets with small doubling.A generalized arithmetic progression (GAP) of rank d in a group G is a set of theform { a + a x + · · · + a d x d | M i ≤ x i ≤ N i } , where a i are elements of G and M i ≤ N i are integers. It is intuitive to view a GAP Q as the image of the d -dimensional integral box B := { ( x , . . . , x d ) | M i ≤ x i ≤ N i } under the linear map Φ( x , . . . , x d ) = a + a x + · · · + a d x d . We say that Q is proper if Φ is one-to-one. It is easy to see that if Q is a properGAP of rank d and A is a subset of density δ of Q , then | A | ≤ C ( d, δ ) | A | . Indeed, | A | ≤ | Q | ≤ | B | = 2 d | B | = 2 d | Q | ≤ d δ | A | since the volume of a box increases by a factor 2 d if its sizes are doubled.Freiman’s theorem shows that this is the only construction of sets with constantdoubling. Theorem 2.1. (Freiman’s theorem) [27]
For any positive constant C , there arepositive constants d = d ( C ) ad δ = δ ( C ) such that the following holds. Let A bea finite subset of a torsion-free group G such that | A | ≤ C | A | . Then there is aproper GAP Q of dimension d such that A ⊂ Q and | A | ≥ δ | Q | . STRUCTURAL APPROACH TO SUBSET-SUM PROBLEMS 5
Freiman theorem has been extended recently to the torsion case by Green andRuzsa [35]. [64, Chapter 5] contains a detailed discussion of both theorems andrelated results.One can use Freiman’s theorem iteratively to treat the sumset lA , for l >
2. Forsimplicity, assume that l = 2 s is a power of 2. Thus, the set A s := lA = 2 s A canbe viewed as 2 A s − where A s − := 2 s − A . Using a multi-scale analysis combinedwith Fremain’s theorem, one can obtain useful structural information about lA or A itself. For an example of this technique, we refer to [61] or [64, Chapter 12].The treatment of l ∗ A and S A is more difficult. However, one can still developstructural theorems in these cases. While the content of most theorems in thisdirection are quite different from that of Freiman’s, they do bear a similar spiritthat somehow the most natural construction happens to be (essentially) the onlyone. 3. Structure of zero-sum-free sets
Let A be a zero-sum-free subset of Z p . We recall the example following Theorem 1.1.Let A := { , , . . . , n } . If 1+ · · · + n < p , then obviously S A does not contain 0. Thisshows that a zero-sum-free set can have close to √ p elements. In [61], Szemer´ediand Vu showed that having elements with small sum is essentially the only reasonfor a set to be zero-sum-free . More quantitative versions of this statement wereworked out in [49] and [50]. For example, we have [50, Theorem 2.2] Theorem 3.1.
After a proper dilation (by some non-zero element), any zero-sum-free subset A of Z p has the form A = A ′ ∪ A ′′ where the elements of A ′ (viewed as integers between 0 and p − ) are small, P x ∈ A ′ x < p , and A ′′ is negligible, | A ′′ | ≤ p / o (1) . One can perhaps improve the constant 6 /
13 by tightening the analysis in [50]. Itis not clear, however, what would be the best constant here. In most applications,it suffices to have any constant strictly less than 1 / A is zero-sum-free (incomplete), then theset A x := { xa | a ∈ A } is also zero-sum-free (incomplete) for any 0 = x ∈ Z p .We can also prove similar results for lA and l ∗ A , and for A being a sequence (see[50] for details). In the rest of this section, we present few applications of theseresults.3.2. The size of the largest zero-sum-free set in Z p . Let m p denote the sizeof the largest zero-sum-free set in Z p . The problem of determining m p was posed by VAN VU
Erd˝os and Heilbronn [23] and has a long history. Szemer´edi proved that m p ≤ C √ p ,for some sufficiently large C independent of p [60]. Olson showed that C = 2 suffices[53]. Much later, Hamidoune and Z´emor [37] showed that m p ≤ √ p + 5 log p ,which is asymptotically sharp. Using an earlier version of Theorem 3.1, Szemer´edi,Nguyen and Vu [49] recently obtained the exact value of m p . Theorem 3.3.
Let n p be the largest integer so that · · · + ( n p − < p . • If p = n p ( n p +1)2 − , then m p = n p − . • If p = n p ( n p +1)2 − , then m p = n p . Furthermore, up to a dilation, the onlyzero-sum-free set with n p elements is {− , , , , . . . , n p } . The same result was obtained by Deshouillers and Prakash (personal communica-tion by Deshouillers) at about the same time.3.4.
The structure of relatively large zero-sum-free sets.
Let us now con-sider the structure of zero-sum-free sets of size close to √ p . Let k x k denote theinteger norm of x . In [15], Deshouillers proved Theorem 3.5.
Let A be a zero-sum-free subset of Z p of size at least √ p . Then(after a proper dilation) X x ∈ A,x
p/ k x/p k ≤ O ( p − / log p ) . Deshouillers showed (by a construction) that the error term p − / cannot be replaceby o ( p − / ). Using an earlier version of Theorem 3.1, Nguyen, Szemer´edi and Vu[49] improved Theorem 4.4 to obtain the best possible error term O ( p − / ), undera stronger assumption on the size of | A | . Theorem 3.6.
Let A be a zero-sum-free subset of Z p of size at least . √ p . Then(after a proper dilation) X x ∈ A,x
p/ k x/p k ≤ O ( p − / ) . The constant .
99 is, of course, ad-hoc and can be improved by redoing the analysiscarefully. On the other hand, it is not clear what the best assumption on | A | shouldbe. STRUCTURAL APPROACH TO SUBSET-SUM PROBLEMS 7
Erd˝os-Ginburg-Ziv revisited.
Using a version of Theorem 3.1 for sequences,Nguyen and Vu [50] obtained the following characterization for a sequence of sizeslightly more than p and does not contain a subsequence of p elements summingup to 0. Corollary 3.8. [50, Theorem 6.2]
Let ε be an arbitrary positive constant. Assumethat A is a p -zero-sum-free sequence and p + p / ǫ ≤ | A | ≤ p − . Then A contains two elements a and b with multiplicities m a , m b satisfying m a + m b ≥ | A | − p − p / ǫ ) . The interesting point here is that the structure kicks in very soon, when A has justslightly more than p elements. Few years ago, Gao, Panigrahi, and Thangdurai [43]proved a similar statement under the stronger assumption that | A | ≥ p/ Incomplete Sets
Now we turn our attention to incomplete sets, namely sets A where S A = Z p . Thesituation here is very similar to that with zero-sum-free sets. Szemer´edi and Vu[61] showed that having elements with small sum is essentially the only reason fora set to be incomplete . More quantitative versions of this statement were workedout in [49] and [50]. For example, in [50], the following analogue of Theorem 3.1was proved Theorem 4.1.
After a proper dilation (by some non-zero element), any incompletesubset A of Z p has the form A = A ′ ∪ A ′′ where the elements of A ′ are small (in the integer norm), P x ∈ A ′ k x/p k < and A ′′ is negligible, | A ′′ | ≤ p / o (1) . The reader can find similar results for lA and l ∗ A and for A being a sequence in[50]. We next discuss some applications of these results.4.2. The structure of relatively large incomplete sets.
Theorem 4.1 enablesus to prove results similar to those in the last section for incomplete sets. Theproblem of determining the largest size of an incomplete set in Z p was first consid-ered by Erd˝os and Heilbronn [23] and essentially solved by Olson (Theorem 1.2).da Silva and Hamidoune [12] tightened the bound to √ p − VAN VU
Concerning the structure of relatively large incomplete sets, Deshouillers and Freiman[17] proved
Theorem 4.3.
Let A be an incomplete subset of Z p of size at least √ p . Then(after a proper dilation) X x ∈ A k x/p k ≤ O ( p − / log p ) . They conjectured that the error term may be replaced by O ( √ p ), which would bebest possible due to a later construction of Deshouillers [16].Using Theorem 4.1, Nguyen and Vu [50] confirmed this conjecture, provided that A is sufficiently close to 2 √ p . Theorem 4.4.
Let A be an incomplete subset of Z p of size at least . √ p . Then(after a proper dilation) X x ∈ A k x/p k ≤ O ( √ p ) . Similar to the constant .
99 (in the previous section), the constant 1 .
99 is ad-hocand can be improved by redoing the analysis carefully. On the other hand, it is notclear what the best assumption on | A | is.4.5. The structure of incomplete sequences.
Let us now discuss (rather briefly)the situation with sequences. The main difference between sets and sequences isthat a sequence can have elements with high multiplicities. It has turned out thatwhen the maximum multiplicity of incomplete sequence A is determined, one canobtain strong structural information about A .Let 1 ≤ m ≤ p be a positive integer and A be an incomplete sequence of Z p withmaximum multiplicity m . Trying to make A as large as possible, we come up withthe following example, B m = {− n [ k ] , ( n − [ m ] , . . . , − [ m ] , [ m ] , [ m ] , . . . , ( n − [ m ] , n [ k ] } where 1 ≤ k ≤ m and n are the unique integers satisfying2 m (1 + 2 + · · · + n −
1) + 2 kn < p ≤ m (1 + 2 + · · · + n −
1) + 2( k + 1) n. It is clear that any subsequence of B m is incomplete and has multiplicity at most m . In [50], we proved that any incomplete sequence A with maximum multiplicity m and cardinality close to | B m | is essentially a subset of B m . STRUCTURAL APPROACH TO SUBSET-SUM PROBLEMS 9
Theorem 4.6.
Let / < α < / be a constant. Assume that A is an incom-plete sequence of Z p with maximum multiplicity m and cardinality | A | = | B m | − O (( pm ) α ) . Then after a proper dilation, we can have A = A ′ ∪ A ′′ where A ′ ⊂ B m and | A ′′ | = O (( pm ) ( α +1 / / ) . Counting problems.
Sometime one would like to count the number of setswhich forbid certain additive configurations. A well-known example of problemsof this type is the Cameron-Erd˝os problem [11], which asked for the number ofsubsets of [ n ] = { , , . . . , n } which does not contain three different elements x, y, z such that x + y − z = 0. Cameron an Erd˝os noticed that any set of odd numbershas this property. Thus, in [ n ] there are at least Ω(2 n/ ) subsets with the requiredproperty. They conjectured that 2 n/ is the right order of magnitude. There wereseveral partial results [2, 9, 22] before Green settled the conjecture [34].Using structural theorems such as Theorem 3.1, one can obtain results of similarspirit for the number of zero-sum-free or incomplete sets and sequences. For exam-ple, using an earlier version of Theorem 3.1 and standard facts from the theory ofpartitions [1], Szemer´edi and Vu [61] proved Corollary 4.8.
The number of incomplete subsets of Z p is exp(( q π + o (1)) √ p ) . Using Theorem 4.6, one obtains the following generalizations [50].
Corollary 4.9.
The number of incomplete sequences A with highest multiplicity m in Z p is exp(( q (1 − m +1 ) π + o (1)) √ p ) . It is an interesting question to determine the error term o (1).5. Incomplete sets in a general abelian group
Let us now consider the problem of finding the largest size of an incomplete set ina general abelian group G , which we denote by In( G ) in the rest of this section.The situation with a general group G is quite different from that with Z p , dueto the existence of non-trivial subgroups. It is clear that any such subgroup isincomplete. Thus, In( G ) ≥ h , where h is the largest non-trivial divisor of | G | . Theintuition behind the discussion in this section is that a large incomplete set shouldbe essentially contained in a proper subgroup .In 1975, Diderrich [13] conjectured that if | G | = ph , where p ≥ | G | and h is composite, then c ( G ) = h + p −
2. (The cases where p = 2 or h is a prime is simpler and were treated earlier, some by Diderrich himself [13, 47, 14].) Didderich’s conjecture was settled by Gao and Hamidoune in 1999[29].The following simple fact explains the appearance of the term p − Fact. If S A ∩ H = H for some maximal subgroup H of (prime) index q , then | A | ≤ | H | + q − A/H is a sequence in the group Z q . It is easyto show (exercise) that if B is a sequence of q − Z q , then S B ∪ { } = Z q .We say that subset A of G is sub-complete if there is a subgroup H of prime indexsuch that S A ∩ H = H .Once we know that an incomplete set A is sub-complete, we can write down itsstructure completely. There is a subgroup H with prime index q such that | A \ H | ≤ q −
2, and the sequence
A/H is incomplete in Z q . (The structure of such a sequencewas discussed in the previous section.) It is natural to pose the following Problem 5.1.
Find the threshold for sub-completeness.
Recently, Gao, Hamidoune, Llad´o and Serra [30] showed (under some weak assump-tion) that any subset of at least pp +2 h + p elements is sub-complete. Furthermore,one can choose H to have index p , where p is the smallest prime divisor of | G | . Vu[68] showed (again under some weak assumption)that h is sufficient to guaranteesub-completeness. It is not clear, however, that what the sharp bound is.The situation is much better if we assume that | G | is sufficiently composite. In par-ticular, if the product of the two smallest prime divisors of | G | is significantly smallerthan p | G | , then one can determine the sharp threshold for sub-completeness. Theorem 5.2. [68]
For any positive constant δ there is a positive constant D ( δ ) such that the following holds. Assume that | G | = p . . . p t , where t ≥ and p ≤ p · · · ≤ p t are primes such that p p ≤ D ( δ ) p | G | / log | G | . Then any incompletesubset A of G with cardinality at least (1 + δ ) np p is subcomplete. Furthermore, thelower bound (1 + δ ) np p cannot be replaced by np p + n / − α , for any constant α . Structures in S A As mentioned in the introduction, an alternative way to implement our plan is tofind a structure in S A rather than in A (Step I’). A well-known result concerningthe structure of S A is the following theorem, proved by Freiman [28] and S´ark¨ozy[55] independently. Theorem 6.1.
There are positive constants C and c such that the following holdsfor all sufficiently large n . Let A be a subset of [ n ] := { , . . . , n } with at least C √ n log n elements. Then S A contains an arithmetic progression of length c | A | . STRUCTURAL APPROACH TO SUBSET-SUM PROBLEMS 11
It is clear that the bound on the length of the arithmetic progression (AP) isoptimal, as one can take A to be an interval. The lower bound on | A | , however,can be improved to C √ n , as showed by Szemer´edi and Vu [62]. Theorem 6.2.
There are positive constants C and c such that the following holdsfor all sufficiently large n . Let A be a subset of [ n ] := { , . . . , n } with at least C √ n elements. Then S A contains an arithmetic progression of length c | A | . The assumption | A | ≥ C √ n is optimal, up to the value of C , as one can constructa set A ⊂ [ n ] of ǫ √ n elements, for some small constant ε , such that S A does notcontain any arithmetic progression of length larger than n / (see [62] or [63, Section3.4]).Theorem 6.2 can be extended considerably. Szemer´edi and Vu [63] showed that forany set A ⊂ [ n ] and any integer l such that l d | A | ≥ n for some constant d , thesumset l ∗ A contains a large proper generalized arithmetic progression (GAP). Theparameters of this GAP is optimal, up to a constant factor (see [63, Section 3] formore details). Theorem 6.3. [63, Theorem 7.1]
For any fixed positive integer d there are positiveconstants C and c depending on d such that the following holds. For any positiveintegers n and l and any set A ⊂ [ n ] satisfying l d | A | ≥ Cn , l ∗ A contains a properGAP of rank d ′ and volume at least cl d ′ | A | , for some integer ≤ d ′ ≤ d . There are variants of Theorem 6.3 for finite fields, and also for sums of different sets(see [63, Section 5] and [63, Section 10]). In the following subsections, we discussfew applications of Theorems 6.2 and 6.3.6.4.
Folkman conjectures on infinite arithmetic progressions.
Let us recallto the conjecture of Folkman, mentioned in the introduction.
Conjecture 6.5. (Folkman’s conjecture) The following holds for any sufficientlylarge constant C . Let A be an strictly increasing sequence of positive integers with(asymptotic) density A ( n ) at least C √ n (namely A ( n ) := | A ∩ [ n ] | ≥ C √ n for allsufficiently large n ). Then S A contains an infinite arithmetic progression. Folkman [25] showed that the conjecture holds under a stronger assumption that A ( n ) ≥ n / ǫ , where ǫ is an arbitrarily small positive constant. (An earlier resultof Erd˝os [20] on a closely related problem can perhaps be adapted to give a weakerbound n ( √ − / .) Hegyv´ari [45] and Luczak and Schoen [46], independently, re-duced the density n ǫ to C √ n log n , using Theorem 6.1.Using the stronger Theorem 6.2, together with some additional arguments, Sze-mer´edi and Vu [62] proved the full conjecture. Theorem 6.6.
Conjecture 6.5 holds.
In the same paper [25], Folkman also made a related conjecture for increasing, butnot strictly increasing sequences. Let A ( n ) now be the number of elements of A (counting multiplicities) at most n . Conjecture 6.7. (Folkman’s second conjecture) The following holds for any suffi-ciently large constant C . Let A be an increasing sequence of positive integers withsuch that A ( n ) ≥ Cn for all sufficiently large n . Then S A contains an infinitearithmetic progression. Despite the huge change from √ n to n in the density bound, this conjecture isalso sharp [25], and (for some time) appeared more subtle than the first one (seea discussion in [21, Chapter 6]). Folkman [25] proved the conjecture under thestronger assumption that A ( n ) ≥ n ε . It does not seem that one can obtain theanalogue of Hegyv´ari and Luczak-Schoen results due to the lack of a ”sequence”variant of Theorem 6.1. However, the method in [63] is sufficiently robust to enableone to obtain such a variant for the stronger Theorem 6.2. With the help of thisresult, one can settle Conjecture 6.7 Theorem 6.8. [63, Section 6]
Conjecture 6.7 holds.
The strategy for the proofs of Theorems 6.6 and 6.8 is the following. We first finda sufficient condition for a sequence A such that S A contains an infinite AP.We say that an infinite sequence A admits a good partition if it can be partitionedinto two subsequences A ′ and A ′′ with the following two properties • There is a number d such that S A ′ contains an arbitrary long arithmeticprogression with difference d . • Let A ′′ = b ≤ b ≤ b ≤ . . . . For any number K , there is an index i ( K )such that P i − j =1 b j ≥ b i + K for all i ≥ i ( K ). Lemma 6.9.
If a sequence A admits a good partition then S A contains an infiniteAP . The second assumption is easy to satisfied given that A has proper density. Thus,the key is the first assumption. The main feature here is that in this assumption,we only need to guarantee the existence of long (but finite) APs. So, Theorem 6.2and its variants can be used with full power to achieve this goal.6.10. Erd˝os conjecture on square-sum-free sets.
In this section, we returnto Erd˝os conjecture on square-sum-free sets, mentioned in the introduction. Let SF ( n ) denote the size of the largest subset A of [ n ] such that S A does not containa square (or A is square-sum-free). Erd˝os [19] observed that SF ( n ) = Ω( n / ) andconjectured that the truth is close to this lower bound. Since then, there havebeen several attempts on his conjecture. Alon [4] proved that SF ( n ) = O ( n log n ) . In[40] Lipkin improved the bound to SF ( n ) = O ( n / ε ) . Later, Alon and Freiman[5] obtained another improvement SF ( n ) = O ( n / ε ) . About fifteen years ago,S´ark¨ozy [56] showed SF ( n ) = O ( √ n log n ) . STRUCTURAL APPROACH TO SUBSET-SUM PROBLEMS 13
Let us now address the problem from our structural approach point of view. The-orem 6.2 is no longer useful, as we are dealing with sets of size around n / , waybelow the lower bound √ n required in this theorem. Fortunately, we have a moregeneral result, Theorem 6.3, which enables us to find structures in S A for any setof size n δ , for any constant δ . In particular, we can deduce from this theorem thefollowing corollary. Corollary 6.11.
There are positive constants C and c such that the following holdsfor all sufficiently large n . Let A be a subset of [ n ] with cardinality at least Cn / .Then S A contains either an AP of length c | A | or a proper GAP of rank andvolume c | A | . Combining this corollary with some number theoretic arguments, Nguyen and theauthor [51] can get close to the conjectured bound.
Theorem 6.12.
There is a constant C such that SF ( n ) ≤ n / log C n . We strongly believe that the log term can be removed. Details will appear elsewhere.7.
Inverse Littlewood-Offord theorems and random matrices
In this final section, we discuss a problem with a slightly different nature. Let A be a sequence of non-zero integers. Now we are going to view S A as a multi-set of2 n elements. We denote by M A be the largest multiplicity in S A . For example, if A = { , . . . , } , then M A = (cid:0) n ⌊ n/ ⌋ (cid:1) = Θ(2 n / √ n ) . The problem of bounding M A originated from Littlewood and Offord’s work onrandom polynomials [41]. In particular, they proved that M A = O (2 n log n/ √ n ) . The log term was removed by Erd˝os [18], who obtained a sharp bound for M A .Many extensions of this result were obtained by various researcher: Erd˝os-Moser[24], S´ark¨ozy-Szemer´edi [58], Katona [38] Kleitman [39], Hal´asz [44], Griggs et. al.[36], Frankl-F¨uredi [26], Stanley [59] etc. Among others, it was showed that thebound on M A keeps improving, if one forbids more and more additive structuresin A . For example, Erd˝os and Moser [24] showed that if the elements of A aredifferent (i.e., A is a set), then M A = O (2 n log n/n / ). In general, the followingcan be deduced from results of [44] (see also [64, Problem 7.2.8]) Theorem 7.1.
For any fixed integer k there is a constant C such that the followingholds. Let A = { a , . . . , a n } and R k be the number of roots of the equation ε a i + · · · + ε k a i k = 0 with ε i = ± and i , . . . , i k ∈ [ n ] . Then M A ≤ Cn − k − / R k . In [66], Tao and Vu introduced the notion of Inverse Littlewood-Offord theorems.The intuition here is that if M A is large (of order 2 n /n C for any constant C , say),then A should have a very strong structure. The most general example we found with large M A is the following. Let Q be aproper GAP of constant rank d and volume V . If A is a subset of Q , it is easy toshow that M A = Ω( n d/ V ) (in order to see this, view the elements of S A as randomsums P ni =1 ξ i a i where a i are elements of A and ξ i are iid random variables takingvalues 0 and 1 with probability 1 / Q is small, then M A is large.In [66], Tao and Vu proved the inverse statement, asserting that having A as asubset of a small GAP is essentially the only way to guarantee make M A large . Theorem 7.2. [66]
For any constant C and ǫ there are constants B and d suchthat the following holds. Let A be a sequence of n elements in a torsion-free group G . If M A ≥ n /n C for some constant C , then all but at most n − ǫ elements of A is contained in a proper GAP Q of rank d and cardinality n B . In a more recent paper [67], the same authors obtained a (near) optimal relationshipbetween the parameters
C, ǫ, d and B . As a corollary, one can deduce (asymptoticversions of) many earlier results, such as Theorem 7.1. (In spirit, this processis somewhat similar to the process of using Theorem 3.1 to reprove, say, Erd˝os-Ginburg-Ziv theorem.)We would like to conclude this survey with a rather unexpected application. Let usleave combinatorial number theory and jump to the (fairy remote) area of mathe-matical physics. In the 1950s, Wigner observed and proved his famous semi-circlelaw concerning the limiting distribution of eigenvalues in a symmetric random ma-trix [69]. A brother of this law, the so-called circular law for non-symmetric randommatrices, has been conjectured, but remains open since that time. Conjecture 7.3. (Circular Law Conjecture) Let ξ be a random variable with mean0 and variance 1 and M n be the random matrix whose entries are iid copies of ξ .Then the limiting distribution of the eigenvalues of √ n M n converges to the uniformdistribution on the unit disk. Girko [31] and Bai [6] obtained important partial results concerning this conjecture.These results and many related results are carefully discussed in the book [7]. Therehas been a series of rapid developments recently by G¨otze-Tikhomirov [32, 33], Pan-Zhou [54], and Tao-Vu [65]. In particular, Tao and Vu [65] confirmed the conjectureunder the slightly stronger assumption that the (2 + η )-moment of ξ is bounded,for any η > Theorem 7.4.
The Circular Law holds (with strong convergence) under an addi-tional assumption that E ( | ξ | η ) < ∞ for some fixed η > . The key element of this proof is a variant of Theorem 7.2, which enables us to countthe number of sequences A with bounded elements such that M A (more preciselya continuous version of it) is large. For details, we refer to [65]. STRUCTURAL APPROACH TO SUBSET-SUM PROBLEMS 15
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