On the Inverse Erdos-Heilbronn Problem for Restricted Set Addition in Finite Groups
aa r X i v : . [ m a t h . C O ] O c t ON THE INVERSE ERD ˝OS-HEILBRONN PROBLEM FORRESTRICTED SET ADDITION IN FINITE GROUPSSuren M. Jayasuriya † Department of Electrical and Computer Engineering, Cornell University, Ithaca,New York 14853, USA [email protected]
Steven D. Reich † Department of Mathematics, The University of Pittsburgh, Pittsburgh,Pennsylvania, 15260, USA [email protected]
Jeffrey P. Wheeler
Department of Mathematics, The University of Pittsburgh, Pittsburgh,Pennsylvania, 15260, USA [email protected]
Abstract
We provide a survey of results concerning both the direct and inverse problems tothe Cauchy-Davenport theorem and Erd˝os-Heilbronn problem in Additive Combina-torics. We formulate a conjecture concerning the inverse Erd˝os-Heilbronn problemin nonabelian groups. We prove an inverse to the Dias da Silva-Hamidoune The-orem to Z /n Z where n is composite, and we generalize this result for nonabeliangroups.
1. Introduction
A basic object in additive combinatorics/additive number theory is the sumset ofsets A and B : † This work was started while S.M. Jayasuriya and S.D. Reich were undergraduates at theUniversity of Pittsburgh in a directed study course supervised by Dr. Jeffrey P. Wheeler.2010
Mathematics Subject Classification : Primary 11P99; Secondary 05E15, 20D60.
Key words and phrases : Cauchy-Davenport Theorem, Erd˝os-Heilbronn Problem, additivenumber theory, sumsets, restricted set addition, finite groups, inverse sumset results, critical pair. efinition 1.1. [Sumset] A + B := { a + b | a ∈ A, b ∈ B } . A simple example of a problem in Additive Number Theory is given two subsets A and B of a set of integers, what facts can we determine about sumset A + B ?One such classic problem was a conjecture of Paul Erd˝os and Hans Heilbronn [12],an open problem for over 30 years until proved in 1994. The conjecture originatesfrom a theorem proved by Cauchy [6] in 1813 and independently by Davenport [8]in 1935.In this paper, we present a survey of results concerning both the Cauchy-Davenporttheorem and Erd˝os-Heilbronn problem . We introduce the two main types of prob-lems: direct and inverse problems, and we then list results and extensions of thesetheorems into groups. In particular, we formulate a conjecture concerning the in-verse Erd˝os-Heilbronn problem into nonabelian groups and provide a nontrivialexample to support it. In section 5, we present an elementary proof providing aninverse to the Dias da Silva-Hamidoune theorem for Z /n Z where n is composite. Insection 6, we generalize this result to nonabelian groups which proves one directionof the conjecture under an assumption that our sets are arithmetic progressionswith the same common difference. This result is a first step towards proving thefull inverse Erd˝os-Heilbronn problem in nonabelian groups.
2. The Cauchy-Davenport Theorem and Erd˝os-Heilbronn Problem
As described by Melvyn B. Nathanson in [28], a direct problem in Additive Num-ber Theory is a problem concerned with properties of the resulting sumset. Wefirst consider two direct results; the first a classic result and the second a simpleadaptation of the first - yet significantly more subtle to prove.
The first result is a theorem proved by Cauchy in 1813 [6] and independently byDavenport in 1935 [8] (Davenport discovered in 1947 [9] that Cauchy had previouslyproved the theorem). In particular, Theorem 2.1. [Cauchy-Davenport]Let A and B be nonempty subsets of Z /p Z with p prime. Then | A + B | ≥ min { p, | A | + | B | − } where A + B := { a + b | a ∈ A and b ∈ B } . We note that in 1935 Inder Cholwa [7] extended the result to composite moduli m when 0 ∈ B and the other members of B are relatively prime to m . See Appendix for a timeline summarizing these results Cauchy used this theorem to prove that Ax + By + C ≡ p ) has solutions providedthat ABC
0. Cauchy then used this to provide a new proof of a lemma Lagrange used toestablish his four squares theorem in 1770 [1]. .2. A Conjecture of Erd˝os-Heilbronn The second result we consider is a slight modification of the Cauchy-DavenportTheorem and is surprisingly much more difficult. In the early 1960’s, Paul Erd˝os andHans Heilbronn conjectured that if the sumset addition is restricted to only distinctelements, then the lower bound is reduced by two. Erd˝os stated this conjectureduring a 1963 University of Colorado number theory conference [12]. While theconjecture did not appear in their 1964 paper on sums of sets of congruence classes[15], Erd˝os lectured on the conjecture (see [28], p.106). The conjecture was formallystated in [13] and [14] as follows:
Theorem 2.2. [Erd˝os-Heilbronn Problem]Let p be a prime and A , B ⊆ Z /p Z with A, B = ∅ . Then | A ˙+ B | ≥ min { p, | A | + | B | − } , where A ˙+ B := { a + b mod p | a ∈ A , b ∈ B and a = b } . The conjecture was first proved for the case A = B by J.A. Dias da Silva andY.O. Hamidoune in 1994 [10] using methods from linear algebra with the moregeneral case (namely A = B ) established by Noga Alon, Melvin B. Nathanson, andImre Z. Ruzsa using their powerful polynomial method in 1995 [2]. Remark 2.3.
Throughout this paper, we will label the Erd˝os-Heilbronn problemfor the case where A = B as the Dias da Silva-Hamidoune theorem.
3. Extension of the Problems to Groups
The structures over which the Cauchy-Davenport Theorem holds have been ex-tended beyond Z /p Z . Before stating the extended versions, the following definitionis needed. Definition 3.1 (Minimal Torsion Element) . Let G be a group. We define p ( G ) tobe the smallest positive integer p for which there exists a nontrivial element g of G with pg = 0 (or, if multiplicative notation is used, g p = 1 ). If no such p exists, wewrite p ( G ) = ∞ . Remark 3.2.
When G is finite, then p ( G ) is the smallest prime factor of | G | orequivalently, p ( G ) is the size of the smallest nontrivial subgroup of G .Equipped with this we can state the Cauchy-Davenport Theorem which wasextended to abelian groups by Kneser [25] and then to all finite groups by Gy.K´arolyi [21]: Theorem 3.3 (Cauchy-Davenport Theorem for Finite Groups) . If A and B arenon-empty subsets of a finite group G , then | AB | ≥ min { p ( G ) , | A | + | B | − } , where AB := { a · b | a ∈ A and b ∈ B } . Theorem 3.4. If A and B are two finite sets in a group, then both | A · B | ≥ | A | + | B | unless A · B · ( − B · B ) = A · B , and there is a subset S ⊆ A · B and subgroup H such that | S | ≥ | A | + | B | − | H | and either H · S = S or S · H = S . Similarly, work has been done to extend the Erd˝os-Heilbronn Problem intogroups. Starting with abelian groups, Gy. K´arolyi proved the following resultin [19, 20]:
Theorem 3.5. If A is a nonempty subset of an abelian group G , then | A ˙+ A | ≥ min { p ( G ) , | A | − } . He also extended the Erd˝os-Heilbronn Problem to cyclic groups of prime poweredorder in 2005 [22].To state the result that extends the problem into finite, not necessarily abelian,groups, we introduce the following definition:
Definition 3.6.
For a group G , let Aut( G ) be the group of automorphisms of G .Suppose θ ∈ Aut( G ) and A, B ⊆ G . Write A θ · B := { a · θ ( b ) | a ∈ A, b ∈ B, and a = b } . Given this definition, we can clearly state the Erd˝os-Heilbronn theorem for finitegroups which was proven in 2009 by P. Balister and J.P. Wheeler [5]:
Theorem 3.7 (Generalized Erd˝os-Heilbronn Problem for Finite Groups) . Let G be a finite group, θ ∈ Aut( G ) , and let A, B ⊆ G with | A | , | B | > . Then | A θ · B | ≥ min { p ( G ) − δ θ , | A | + | B | − } where A θ · B := { a · θ ( b ) | a ∈ A, b ∈ B, and a = b } and where δ θ = (cid:26) if θ has even order in Aut( G ) , if θ has odd order in Aut( G ) . We note that Lev [27] has shown that Theorem 3.7 does not hold in general foran arbitrary bijection θ .
4. Inverse Problems
The previous problems were concerned with properties of the sumset given someknowledge of the individual sets making up the sumset. This leads one to considerquestions in the other direction. In particular, if we know something about thesumset, does this give us any information about the individual sets making up thesumset? Again we borrow the language of [28] and refer to these problems as inverseproblems . In each case, the inverse of the previously stated problems yields beautifulresults. 4 .1. Inverse Problems for the Cauchy-Davenport Theorem
The first result is the inverse to the Cauchy-Davenport theorem in Z /p Z due toA.G. Vosper [31]: Theorem 4.1 (Vosper’s Inverse Theorem) . Let A and B be finite nonempty subsets of Z /p Z . Then | A + B | = | A | + | B | − ifand only if one of the following cases holds: ( i ) | A | = 1 or | B | = 1 ; ( ii ) A + B = Z /p Z ; ( iii ) | A + B | = p − and B is the complement of the set c − A in Z /p Z where { c } = ( Z /p Z ) \ ( A + B ) ; ( iv ) A and B are arithmetic progressions of the same common difference.
Remark 4.2.
We note that when | A + B | = | A | + | B | −
1, we label sets A and B as a critical pair .In 1960, J.H.B. Kemperman [24] extended a weaker version of Vosper’s Theoremto abelian groups. Namely, Theorem 4.3 (Kemperman) . Let A be a nonempty subset of an abelian group. Let p ( G ) be as in Definition 3.1.Suppose p ( G ) > | A | − . Then | A + A | = 2 | A | − if and only if A is an arithmeticprogression. To generalize the inverse problem in groups, we introduce the notion of an arith-metic progression in a group:
Definition 4.4 (Group Arithmetic Progression) . Let G be a group and A ⊆ G with | A | = k . Then A is a group arithmetic progression if there exists both g and h in G such that A = { g + ih | ≤ i < k } . In the above definition, we say that A is a k -term group arithmetic progressionwith common difference h . If the group G is nonabelian, we utilize multiplicativenotation and form the intuitive definitions of left and right arithmetic progressions.Y.O. Hamidoune further extended this idea to finitely generated groups [17]. Theorem 4.5 (Hamidoune) . Let G be a (not necessarily abelian) group generated by a finite subset S where ∈ S .Then either1. for every subset T such that ≤ | T | < ∞ , we have | S + T | ≥ min {| G | − , | S | + | T |} or2. S is an arithmetic progression. Theorem 4.6.
Let
A, B be subsets of a finite group G such that | A | = k , | B | = l ,and k + l − ≤ p ( G ) − . Then | AB | = k + l − where AB = { ab | a ∈ A, b ∈ B } if and only if one of the following conditions holds: ( i ) k = 1 or l = 1 ; ( ii ) there exists a, b, q ∈ G such that A = { a, aq, aq , . . . , aq k − } and B = { b, qb, q b, . . . , q l − b } ;( iii ) k + l − p ( G ) − and there exists a subgroup F of G of order p ( G ) and elements u, v ∈ G, z ∈ F such that A ⊂ uF, B ⊂ F v and A = u ( F \ zvB − ) . We as well have inverse problems for the Erd˝os-Heilbronn Problem. More specifi-cally, we have inverse results for the Dias da Silva-Hamidoune Theorem. In partic-ular, Gy. K´arolyi established the following in [22]:
Theorem 4.7 (Inverse Theorem of the Dias da Silva-Hamidoune Theorem) . Let A be a subset of Z /p Z where p is a prime. Further suppose | A | ≥ and p > | A | − .Then | A ˙+ A | = 2 | A | − if and only if A is an arithmetic progression. He also extended this result to the following:
Theorem 4.8.
Let A be a subset of an abelian group G where p ( G ) is as in Definition 3.1 prime.Further suppose | A | ≥ and p ( G ) > | A | − . Then | A ˙+ A | = 2 | A | − if and onlyif A is a group arithmetic progression. Similar to the inverse to the Cauchy-Davenport theorem which holds in non-abelian groups, Gy. K´arolyi conjectured that the inverse to the Erd˝os-Heilbronnproblem in a nonabelian setting should hold, namely that only sets that are arith-metic progressions achieve the lower bound placed on their restricted sumset by theErd˝os-Heilbronn problem [23]. We note that previous work by V. F. Lev proved aninverse Erd˝os-Heilbronn theorem in an asymptotic sense for Z /p Z with p very large[26], and that this result was improved by Van Vu and Philip M. Wood [32]. Duand Pan [11] have recently submitted a proof for the following result: Theorem 4.9.
Suppose that
A, B are two non-empty subsets of the finite nilpotentgroup G . If A = B , then the cardinality of A ι · B is at least the minimum of p ( G ) and | A | + | B | − . | A ˙+ A | = 2 | A | − A a non-empty subset of a finitegroup G with | A | < ( p ( G ) + 3) /
2, then A is commutative.Thus we formulate a conjecture for the inverse Erd˝os-Heilbronn problem to holdin groups that are not nilpotent. We only consider restricted product sets with θ = ι being the identity automorphism. Conjecture 4.10.
Let
A, B be nonempty subsets of a finite (not necessarily abelian),non-nilpotent group G where p ( G ) is as in Definition 3.1. Further suppose | A | = k ≥ , | B | = l ≥ , and k + l − < p ( G ) . Then | A ι · B | = | A | + | B | − where A ι · B = { ab | a ∈ A, b ∈ B, a = b } if and only if there exists a, q ∈ G such that A = { a, aq, aq , . . . , aq k − } and B = { a, qa, q a, . . . , q l − a } where aq k − = q l − a , i.e. A, B share the same endpoints.
We note that the if statement of the conjecture is trivial, and if G is nilpotent,such pairs A and B only exist when A = B is a progression lying in an abeliansubgroup as shown in [11].We now present an example of sumset addition in a non-nilpotent group thatgives evidence that extra critical pairs can indeed arise. The difficulty in testing thisconjecture is finding an appropriate group whose p ( G ) is relatively large comparedto the cardinalities of sets A and B . Standard nonabelian groups such as dihedralgroups or symmetric groups do not satisfy this condition because p ( G ) = 2 in thesegroups. Thus in the following example, we construct a large (in terms of p ( G ))nonnilpotent group to test the conjecture. Note: since writing this manuscript, wehave discovered simpler and more general examples that will be presented in futurework [18]. Example 4.11.
Let G be a nonabelian group constructed by G = ( Z × Z ) ⋊ φ Z . Since
Aut ( Z × Z ) ∼ = GL ( F ), we construct the homomorphism φ : Z → GL ( F )as follows: φ ( x ) = (cid:18) x
00 1 (cid:19) . Explicitly, we can think of elements of G having the form (cid:18)(cid:18) xy (cid:19) , z (cid:19) wherethe group operation is (cid:18)(cid:18) xy (cid:19) , z (cid:19) · G (cid:18)(cid:18) x ′ y ′ (cid:19) , z ′ (cid:19) = (cid:18)(cid:18) xy (cid:19) + φ ( z ) (cid:18) x ′ y ′ (cid:19) , z + z ′ (cid:19) . = (cid:18)(cid:18) x + 2 z x ′ y + y ′ (cid:19) , z + z ′ (cid:19) . We denote Z / Z as Z for notational purposes, and note that Z ∼ = F +47 . p ( G ) = 23 since | G | = 23 · . Take A = ((cid:18)(cid:18) (cid:19) , (cid:19) · (cid:18)(cid:18) (cid:19) , (cid:19) k (cid:12)(cid:12) ≤ k ≤ ) = (cid:26)(cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19)(cid:27) . and let B = ((cid:18)(cid:18) (cid:19) , (cid:19) l · (cid:18)(cid:18) (cid:19) , (cid:19) (cid:12)(cid:12) ≤ l ≤ ) = (cid:26)(cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19)(cid:27) . So A is right arithmetic progression with cardinality | A | = 5 and B is a leftarithmetic progression with cardinality | B | = 9. Further, A and B share the sameendpoints and have the same “common difference” q . Explicitly computing A ι · B ,we get 11 elements, which is equal to | A | + | B | − A ι · B = (cid:26)(cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19) , (cid:18)(cid:18) (cid:19) , (cid:19)(cid:27) . In the following two sections, we prove a series of results that prove the forwarddirection of this conjecture under the assumption that we have a priori knowledgethat A and B are arithmetic progressions with the same common difference.
5. An Extension to Z /n Z for the Inverse Theorem of the Dias da Silva-Hamidoune Theorem Our first result extends Theorem 4.7 in Z /n Z for composite n by assuming we have apriori knowledge of the sets A, B as arithmetic progressions with the same commondifference, and characterizing when such
A, B form a critical pair, i.e. reach thelower bound of the Erd˝os-Heilbronn Problem.
Theorem 5.1.
Let
A, B ⊆ G = Z /n Z where | A | = k, | B | = l and p ( G ) is thesmallest prime dividing n. Suppose p ( G ) > k + l − where k, l ≥ . Further supposethat
A, B are arithmetic progressions with the same common difference. Then wehave that: | A · + B | = | A | + | B | − implies A = B. roof. Let d be the common difference of the arithmetic progressions A and B , i.e. A = { a + sd | ≤ s ≤ k − } and B = { b + td | ≤ t ≤ l − } . Without loss ofgenerality, we can suppose that | A | ≥ | B | , i.e. k ≥ l .We have that A · + B = { a + sd + b + td | ≤ s ≤ k − , ≤ t ≤ l − , a + sd = b + td } . Since a + sd + b + td = a + ( s ± d + b + ( t ∓ d, (1)then even if a + sd = b + td , by the above we have that the sum can still be writtenas the sum of two distinct elements from A and B unless(i) s = t = 0 or(ii) s = k − t = l − . In other words, we can find another pair of elements, a +( s ± d ∈ A , and b +( t ∓ d ∈ B , that yield the same sum, unless the term in question is a shared endpointof the arithmetic progression where s, t = 0 corresponds to the first endpoint and s = k − , t = l − A · + B = A + B ( ⇒ | A · + B | ≥ | A | + | B | −
1, contrary to our assumption),unless:(i) a = b or(ii) a + ( k − d = b + ( l − d Notice without loss of generality that (ii) can be reduced to (i) by putting ¯ a = a + ( k − d, ¯ b = b + ( l − d and forming the arithmetic progressions by setting¯ d = − d .Now since a = b , we have that A · + B = { a + sd + a + td | sd = td } . Notice if sd = td for s = t (say without loss of generality that s > t ), then ( s − t ) d ≡ Z /n Z , which implies that n | ( s − t ) d . Because d
0, there is a prime p dividing n such that p | ( s − t ). By our definition of p ( G ) as the smallest prime dividing n ,we have k ≥ ( s − t ) ≥ p ≥ p ( G ) > k + l − ≥ k + 3 − k which is a contradiction (Note: if we had assumed t > s , we would have derived asimilar contradiction using l ). Thus we must conclude that sd = td implies s = t .Again, we now point out that if sd = td , then we can write a + sd + a + td = a + ( s ± d + a + ( t ∓ d s, t = 0 or s = t = k − l − k > l ,then we only get the case s = t = 0 which means that | A · + B | ≥ | A + B \ { a + a }| ≥ | A | + | B | − − | A | + | B | − . This is a contradiction to our assumption, so we are forced to conclude that k = l ,and so then A = B . This completes the proof. Corollary 5.2.
Let
A, B ⊆ Z /n Z be arithmetic progressions with the same commondifference where | A | = k, | B | = l . Suppose p > k + l − where k, l ≥ . Then wehave that: | A · + B | = | A | + | B | − if and only if A = B. Proof.
The forward direction is a consequence of Theorem 5.1. The converse is aspecial case of Theorem 4.6 where the abelian group is Z /n Z . Remark 5.3.
As pointed out to us by Gy. K´arolyi, the assumption that A and B are arithmeticprogressions can be dropped when we are in Z /p Z where p is prime to yield thefollowing result [23]: Theorem 5.4.
Let
A, B ⊆ Z /p Z be nonempty subsets such that p ≥ | A | + | B | − .Then | A · + B | = | A | + | B | − if and only if A = B and one of the following holds: ( i ) | A | = 2 or | A | = 3 ; ( ii ) | A | = 4 , and A = { a, a + d, c, c + d } ; ( iii ) | A | ≥ , and A is an arithmetic progression. We note that the proof of this statement relies on the polynomial method ofAlon, Nathanson, and Rusza, while our proof relies solely on elementary methodsalbeit with the additional a priori knowledge of
A, B being arithmetic progressionsto prove the result for general Z /n Z . It is not clear to us whether the methodsused to prove Theorem 5.4 can be easily applied to prove Theorem 5.1, but we alsopresent our elementary proof to foreshadow methods used to extend this result intononabelian groups in the next section.
6. A Generalization of the Inverse Theorem of the Dias da Silva-HamidouneTheorem to Nonabelian Groups
In this section, we extend the results of Theorem 5.1 to nonabelian groups when theautomorphism θ = ι is the identity map so that A ι · B = { ab | a ∈ A, b ∈ B, a = b } .10 heorem 6.1. Let
A, B ⊆ G , where | A | = k, | B | = l and p ( G ) is the smallestprime dividing the order of Gy. Suppose p ( G ) > k + l − where k, l ≥ . Furthersuppose that A is a right geometric progression and B is a left geometric progressionand that they have the same common ratio. Then we have that: | A ι · B | = | A | + | B | − implies A and B have the same endpoints . Proof.
Let d be the common ratio of the geometric progressions A and B , i.e. A = { ad s | ≤ s ≤ k − } and B = { d t b | ≤ t ≤ l − } . So we see A ι · B = { ad s d t b | ≤ s ≤ k − , ≤ t ≤ l − , ad s = d t b } . We note that ad s d t b = ad s ± d t ∓ b . Subcase 1:
First suppose that there is at least one pair of elements that cannot be rewritten,i.e. that there exists s, t such that ad s = d t b and ad s +1 = d t − b . Then we see thatwe have d t − b = d t bd which implies b = dbd and similarly ad s +1 = dad s implies that a = dad . Looking at the unrestricted productset, we see that AB = { ad r b | ≤ r ≤ k + l − } and since from our initial assumption | d | ≥ p ( G ) > k + l − | AB | = k + l −
1. So we see the only way the order of the product can achievethe lower bound is to have two or more pairs of elements that are equal. In otherwords, our assumption requires the existence of s , t , s , t such that ad s = d t b , ad s = d t b , with s + t = s + t . Using the identities a = dad and b = dbd from above, we obtain d ( s + t ) − ( s + t ) = 1, from which it follows that | d | divides( s + t ) − ( s + t ). Hence k + l − < p ( G ) ≤ | d | ≤ ( s + t ) − ( s + t ) ≤ k + l − Subcase 2:
We are now reduced to the case when ad s = d t b always implies ad s ± = d t ∓ b forall s, t . Then for each instance of restriction, we can find another pair of elements, ad s ± ∈ A , and d t ∓ b ∈ B , that yield the same product, unless the term in questionis a shared endpoint of the geometric progression where s, t = 0 corresponds to thefirst endpoint and s = k − , t = l − A ι · B = AB = { ad r b | ≤ r ≤ k + l − } (and from our initial assumptions | d | > k + l − | A ι · B | = | AB | = k + l − a = b or ad k − = d l − b .Notice without loss of generality that the second case can be reduced to the firstby putting ¯ a = ad k − , ¯ b = d l − b and forming the geometric progressions by setting¯ d = d − .So we can assume a = b . Now suppose ad k − = d l − b . Then A ι · B = AB \ { ab } implies that | A ι · B | = k + l −
2, which contradicts the initial assumption.Hence we have shown for both subcases that we reach a contradiction. Therefore,we are forced to conclude that A and B must share both endpoints.11 emark 6.2. As opposed to the abelian case, we note that A and B can share thesame endpoints and still not have the same cardinalities as shown in Example 4.11.
7. Current Progress on the Full Conjecture
We note in this section our most recent result whose proof will be presented inanother manuscript [18] that is the most general statement we can prove towardsthe conjecture:
Theorem 7.1.
Let
A, B be subsets of a finite group G such that k = | A | , l = | B | > and p ( G ) > (2 k + 2 l ) k + l , and let σ ∈ Aut(G). If | A σ · B | = k + l − , then thereexist a, q, r ∈ G such that σ ( r ) = q, aq k − = p l − a and A = { a, aq, . . . , aq k − } , B = { a, ra, . . . , r l − a } .
8. Concluding Remarks
We have provided a survey of results concerning both direct and inverse problemsrelated to the Cauchy-Davenport and Erd˝os-Heilbronn problems. We formulatedan open conjecture concerning the inverse Erd˝os - Heilbronn problem in nonabeliangroups and provided a nontrivial group as an example to support our formulation.We proved an inverse theorem of the Dias da Silva-Hamidoune theorem in Z /n Z for composite n under the assumption that A, B are arithmetic progressions ofthe same common difference. While this result may be deducible from methodsused to prove Theorem 5.4 (it is not immediate to us whether this theorem caneasily be generalized to Z /n Z ), we present a novel proof using only elementarymethods. Further, this proof foreshadows a similar argument to extend the resultinto nonabelian groups for the restricted product set with identity automorphism A ι · B = { ab | a ∈ A, b ∈ B, a = b } .Further research includes trying to settle Conjecture 4.10. The example of acritical pair presented in Section 4 has led to recent discoveries of other criticalpairs, and we state our latest result in Section 7 that will be presented in [18]. Thefull conjecture still eludes us, and it is unclear if our elementary methods can beutilized further in this domain. We hope a promising line of attack involving thepolynomial method can be developed for nonabelian groups, and that the inverseErd˝os-Heilbronn problem with arbitrary automorphism θ can be fully establishedin an elegant manner. 12 cknowledgements The authors are very deeply indebted to Gyula K´arolyi for invaluable insight anddiscussion concerning the conjecture of the inverse Erd˝os-Heilbronn problem in non-abelian groups. The authors also wish to thank Bill Layton for his advice andguidance concerning this manuscript. 13 . Appendix
Table 1:
Timeline for Cauchy-Davenport Theorem and Erd˝os-Heilbronn Problem
Year Contents Person(s) Cite1813 Cauchy-Davenport Theorem for Z /p Z Cauchy, A.L. [6]1935 Cauchy-Davenport Theorem for Z /p Z Davenport, H. [8]1935 Cauchy-Davenport Theorem for Z /m Z Chowla, I. [7]1947 acknowledged Cauchy’s work Davenport, H. [9]1953 CDT extended to abelian groups Kneser, M. [25]early developed the Erd˝os-Heilbronn conjecture Erd˝os, P.1960’s Heilbronn, H.1963 stated EHP at Number Theory conference Erd˝os, P. [12]1964 EHP in paper on sumsets of congruence classes Erd˝os, P. [15]Heilbronn, H.1971 EHP appeared in the book Erd˝os, P. [13]
Some Problems in Number Theory
Old and New Problems and
Erd˝os, P. [14]
Results in Combinatorial Number Theory
Graham, R.1984 CDT for finite groups proven Olson, J.E. [29](special case of Olson’s theorem)1994 EHP proved for special case A = B Dias da Silva, J.A. [10]Hamidounne, Y.O.Alon, Noga1995 EHP proved by the Polynomial Method Nathanson, M.B. [3]Ruzsa, I.2000 proved inverse EHP in the asymptotic sense for Z /p Z Lev, V.F. [26]2003 inverse CDT extended to abelian groups K´arolyi, Gy. [21]2004 EHP to abelian groups for A = B K´arolyi, Gy. [19, 20]2004 EHP to groups of prime power order K´arolyi, Gy. [20]2005 / Z /p Z Vu, V. [32]Wood, Philip M.2012 submitted proof that all critical pairs in a finite Du, S. [11]nilpotent group G are of the form A = B Pan, H.KEY: CDT = Cauchy-Davenport Theorem, EHP = Erd˝os-Heilbronn Problem14 eferences [1] N. Alon,
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