Rectification Principles in Additive Number Theory

Discrete & Computational Geometry, Mar 1998

Abstract. We consider two general principles which allow us to reduce certain additive problems for residue classes modulo a prime to the corresponding problems for integers. <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p343.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader>

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Rectification Principles in Additive Number Theory

Discrete Comput Geom Y. F. Bilu 2 3 V. F. Lev 1 3 I. Z. Ruzsa 0 3 0 Mathematical Institute of the Hungarian Academy of Sciences , Pf. 127, H-1364 Budapest , Hungary 1 Department of Mathematics, University of Georgia , Athens, GA 30602 , USA 2 Forschungsinstitut fu ̈r Mathematik, ETH-Zentrum , CH-8092 Zurich , Switzerland 3 Rectification Principles in Additive Number Theory We consider two general principles which allow us to reduce certain additive problems for residue classes modulo a prime to the corresponding problems for integers. ¤ Research of the first author was supported by the SFB 170 “Geometrie und Analysis” at Go¨ttingen. Research of the third author was supported by Hungarian National Foundation for Scientific Research, Grant No. 17433. - It is well known that additive problems in finite abelian groups are generally more difficult than analogous problems in Z. For instance, consider the following classical problem: given an abelian group G, describe all pairs of finite sets K ; L µ G such that jK C L j < jK j C jL j: When G D Z (or a torsion-free abelian group) the answer is almost trivial: K and L must be arithmetic progressions with the same difference. When G is a cyclic group of prime order, the answer is given by Vosper’s theorem [ 24 ], which is quite nontrivial. And when G is an arbitrary finite abelian group, we should turn to the extremely complicated recursive classification of Kemperman [ 14 ]. (A few years ago the problem was solved for torsion-free nonabelian groups [ 6 ], [ 13 ].) Nevertheless, more than 30 years ago, Freiman [10, Sect. 3.13] discovered that, at least for cyclic groups of prime order, certain additive problems can be reduced to corresponding problems in Z, provided the sets in consideration are “not very large.” Seemingly, this important observation did not receive much attention. On the other hand, it was recently observed in [ 15 ] that there is another reduction method. It is weaker than Freiman’s method in that it requires the sets to be “very small” instead of “not very large.” However, unlike the method of Freiman it imposes no additional restrictions on the sets and handles easily the case of distinct summands, which makes it applicable when Freiman’s method fails. The objective of this paper is to apply Freiman’s discovery and the reduction method mentioned above to concrete additive problems. 2. Freiman’s Rectification Principle For simplicity, we consider only the case of equal summands: K D L. Using [22, Lemma 3.3] we can extend the results to distinct summands. We need the concept of Fs -isomorphism [ 10 ], [ 22 ]. Let G; H be abelian groups, and consider subsets K ½ G and L ½ H. The bijection ': K ! L is Freiman’s isomorphism of order s or, shortly, Fs -isomorphism, if for any a1; : : : ; a2s 2 K , a1 C ¢ ¢ ¢ C as D asC1 C ¢ ¢ ¢ C a2s if and only if '.a1/ C ¢ ¢ ¢ C '.as / D '.asC1/ C ¢ ¢ ¢ C '.a2s /: It is easily seen that, if K and L are FsC1-isomorphic, then they are also Fs -isomorphic. Clearly, two sets K and L are F1-isomorphic if and only if jK j D jLj. Theorem 2.1 (Freiman’s Rectification Principle). For any positive numbers ¾ 2 R and s 2 Z there exists a positive constant c1 D c1.¾; s/ such that the following holds: Let p be a prime number and let K µ Z= pZ satisfy jK j · c1 p and jK C K j < ¾ jK j: Then there exists a set of integers K 0 ½ Z such that the canonical homomorphism Z ! Z= pZ induces an Fs -isomorphism of K 0 onto K . To put it briefly, this theorem asserts that if a set of residues has a small sumset and is not too large itself, then it is Fs -isomorphic to a set of integers. A proof of Theorem 2.1 (for s D 2) is briefly sketched in [10, Sect. 3.12]. Our proof given below is simpler than the original, but requires substantially no new ideas. The argument is based on the following result of Freiman: Theorem 2.2 (Freiman). Let ¾ be a positive real number and let K be a finite set of integers satisfying jK C K j < ¾ jK j: Then there exist positive integers r; b1; : : : ; br and nonzero integers g0; g1; : : : ; gr such that r · c2.¾ /; b1 ¢ ¢ ¢ br · c3.¾ /jK j; and K µ 5 D 5.g0I g1; : : : ; gr I b1; : : : ; br / :D fg0 C ¯1g1 C ¢ ¢ ¢ C ¯r gr : ¯i D 0; : : : ; bi ¡ 1g; where c2.¾ / and c3.¾ / are positive constants, depending only on ¾ . There are two different proofs of this theorem. The first is Freiman’s original, see [ 10 ], [ 11 ], and [ 3 ]. The second is due to Ruzsa, see [ 23 ] and [ 20 ]. Proof of Theorem 2.1. We shall see that the theorem holds with c1 D .2sc2.2¾ //¡c2.2¾ /.c3.2¾ //¡1: Let K0 µ f0; 1; : : : ; p ¡ 1g be the preimage of K under the canonical homomorphism Z ! Z= pZ. Then jK0 C K0j · 2jK C K j < 2¾ jK0j, whence by Theorem 2.2 we have K0 µ 5.g0I g1; : : : ; gr I b1; : : : ; br /, where r; b1; : : : ; br are positive integers satisfying r · c2.2¾ /; b1 ¢ ¢ ¢ br · c3.2¾ /jK j; and g1; : : : ; gr 2 Z. Put "i D .2sr bi /¡1. Then p"1 ¢ ¢ ¢ "r D p.2sr /¡r .b1 ¢ ¢ ¢ br /¡1 ¸ p.2sc2.2¾ //¡c2.2¾ /.c3.2¾ //¡1jK j¡1 D c1 pjK j¡1 ¸ 1: Hence by Minkowski’s theorem on linear inequalities [7, App. B, Theorem III], there exists a nonzero vector .a; a1; : : : ;ar / 2 ZrC1 such that ¯¯ agi ¯¯ p jaj < p; ¯ ¡ ai ¯¯ · "i ¯ where k ¢ k stands for the distance from the nearest integer. This gives °° a.x ¡ g0/ °° °° p °° · r ° agi °° X bi °°° p °° · .2s/¡1 iD1 for any x 2 K0. Let K1 µ ½ ¹ p º ¡ .2s/ ; ¡ ¹ p º .2s/ C 1; : : : ; ¹ p º¾ .2s/ be the set of integers congruent modulo p to one of the numbers a.x ¡ g0/ (for some x 2 K0). Clearly, K1 is mapped onto K by x 7! ux C g0 mod p, where u is an arbitrary integer satisfying au ´ 1 .mod p/. It follows that K1 is Fs -isomorphic to K , for any algebraic sum of 2s elements of K1 which is 0 modulo p is also 0 in Z. Finally, we define K 0 D fux C g0: x 2 K1g. Freiman’s rectification principle allows us to reduce various additive problems in Z= pZ to corresponding problems in Z. Unfortunately, we have to make the restrictive assumption jK j · c1 p. Restricting to “not very large” sets is the price we have to pay for the use of such a powerful tool as Freiman’s Theorem 2.2. For instance, the main results of [ 4 ] and [ 5 ] also assume that the sets in question are small enough. It would be nice to find a proof of the rectification principle independent of Theorem 2.2. For ¾ < 2:4 and s D 2 such a proof is implicit in [10, Sect. 2.3], where the following result is obtained. Theorem 2.3 [ 10 ]. Let K µ Z= pZ satisfy jK CK j < 2:4jK j, and suppose, in addition, that jK j < p=35. Then K can be covered by a “short” arithmetic progression modulo p: There exist g0; g1 2 Z= pZ and a positive integer b · jK C K j ¡ jK j C 1 such that K µ fg0 C g1¯: ¯ D 0; : : : ;b ¡ 1g: .2:2/ Conjecturally, the assertion of this theorem is true for jK CK j · maxf p¡1; 3jK j¡4g. Using Theorem 2.1, we can easily prove it for sufficiently small K . Indeed, suppose that jK j < c1 p, where c1 D c1.3; 2/ is the constant of Theorem 2.1. Let K 0 ½ Z be the set of integers which is F2-isomorphic to K and is mapped onto K by the canonical mapping Z ! Z= pZ. Then K 0 also satisfies jK 0 C K 0j · 3jK 0j ¡ 4 (since jK 0j D jK j and jK 0 C K 0j D jK C K j by the definition of an Fs -isomorphism). By another and well-known result of Freiman [10, Theorem 1.9] (which has a relatively easy elementary proof; see also [ 17 ]), there exist a; d 2 Z and a positive integer b · jK 0 C K 0j ¡ jK 0j C 1 such that K 0 µ fa C d¯: ¯ D 0; : : : ;b ¡ 1g: Now (2.2) is clearly satisfied if g0 and g1 are the elements of Z= pZ congruent to a and d, respectively. This is a first illustrative example which shows how rectification methods can be used to reduce difficult additive problems in Z= pZ to easier problems in Z. 3. Direct Rectification Theorem 2.1 shows that any (not too large) set of residues K µ Z= pZ with a small sumset is isomorphic to a set of integers. It turns out that this is true for any K , regardless of its sumset, provided that jK j is very small. Specifically, it is shown in [ 15 ] that any K µ Z= pZ of the cardinality k D jK j is contained in an arithmetic progression modulo p of at most terms. If k · log4 p C log4 log4 p (where log4 is the logarithm base 4), then the number (3.1) is less than p=2 C 1, whence K is F2-isomorphic to a set of integers. Similarly, if k · log2s p C log2s log2s p, then K is Fs -isomorphic to a set of integers. Essentially the same can be obtained by a direct application of the idea we used in the proof of Theorem 2.1. Theorem 3.1. Let K µ Z= pZ, where p is a prime. If jK j · log2s p, then there exists a set of integers K 0 ½ Z such that the canonical homomorphism Z ! Z= pZ induces an Fs -isomorphism of K 0 onto K . Proof. Let K D fg1; : : : ; gr g, and put "1 D ¢ ¢ ¢ D "r D .2s/¡1. Since r · log2s p, we have p"1 ¢ ¢ ¢ "r ¸ 1. Therefore, applying Minkowski’s theorem exactly in the same way as in the proof of Theorem 2.1, we find a 2 Z satisfying .3:1/ a 6´ 0 .mod p/; °° agi °° °° p °° · .2s/¡1 Now let mi 2 f¡b p=.2s/c; ¡b p=.2s/c C 1; : : : ; b p=.2s/cg be defined from mi ´ agi .mod p/, and put K 0 D fum1; : : : ; umk g where u is any integer, inverse to a in Z= pZ. Then the canonical homomorphism Z ! Z= pZ maps K 0 onto K , and this mapping is an Fs -isomorphism, which follows immediately from jmi j < p=.2s/ (as in the proof of Theorem 2.1). This theorem is nearly best possible: here is an example which shows that there exists a set K µ Z= pZ of cardinality jK j · 2 log2 p C 1 which is not F2-isomorphic to any set of integers. (This example can easily be generalized to produce a set of cardinality at most 2 logs p C 1 which is not Fs -isomorphic to any set of integers.) Put N D blog2 pc, and write p D 2d1 C 2d2 C ¢ ¢ ¢ C 2dt , where 0 · d1 < d2 < ¢ ¢ ¢ < dt ; t · N C 1. We define K D f0g [ f1; 2; 4; : : : ; 2N g [ f2d1 C 2d2 ; 2d1 C 2d2 C 2d3 ; : : : ; 2d1 C 2d2 C ¢ ¢ ¢ C 2dt¡1 g (all the numbers are considered as residues modulo p), so that jK j · 2N C 1. We assume that K is F2-isomorphic to a set of integers K 0 D f0g [ fa0; a1; a2; : : : ; aN g [ fb2; b3; : : : ; bt¡1g implies Next, from we obtain and then from which yields subsequently and obtain a contradiction. Let ': K ! K 0 be the isomorphism. As the notation suggests, we suppose (which does not restrict the generality) that '.0/ D 0 and that ai ; bi are the images in K 0 of the corresponding elements of K . Then for any i 2 [0; N ¡ 1], the equality in Z= pZ 0 C 2iC1 D 2i C 2i 0 C aiC1 D ai C ai ; for i D 2; : : : ; t ¡ 2 we obtain which yields But this is a contradiction in view of a1 D 2a0; a2 D 4a0 ; : : : ; aN D 2N a0: 0 C .2d1 C 2d2 / D 2d1 C 2d2 b2 D .2d1 C 2d2 /a0; 0 C .2d1 C ¢ ¢ ¢ C 2di C 2diC1 / D .2d1 C ¢ ¢ ¢ C 2di / C 2diC1 biC1 D bi C adiC1 ; b3 D .2d1 C 2d2 C 2d3 /a0; : : : ; bt¡1 D .2d1 C 2d2 C ¢ ¢ ¢ C 2dt¡1 /a0: To show how Theorem 3.1 can be applied in the case of distinct summands, consider the following problem: Given s sets K1; : : : ; Ks in an abelian group G, how many representations of the form x D a1 C ¢ ¢ ¢ C as I ai 2 Ki .i D 1; : : : ; s/ .3:2/ can an element x 2 G have? We assume here that the cardinalities jKi j are preassigned. For G D Z, we have the following result: Theorem 3.2 [16, Theorem 1]. Let K1; : : : ; Ks µ Z be a finite sets of integers. Then the number of solutions of (3.2) is maximized, when x D 0, and Ki are the sets of consecutive integers Ki D f®i ; ®i C 1; : : : ; °i g .i D 1; : : : ; s/; where integers ®i ; °i are chosen in such a way that °i ¡ ®i C 1 D jKi j; j®i C °i j · 1 .i D 1; : : : ; s/; j.®1 C °1/ C ¢ ¢ ¢ C .®s C °s /j · 1: Using direct rectification we can easily transfer Theorem 3.2 to small subsets of Z= pZ. Theorem 3.3. Let K1; : : : ; Ks µ Z= pZ be sets of residues modulo a prime p, and assume that jK1j C ¢ ¢ ¢ C jKs j · log2s p. Then the number of solutions of (3.2) is maximized, when x D 0, and Ki are the sets of consecutive residues Ki D f®i ; ®i C 1; : : : ; °i g .mod p/ .i D 1; : : : ; s/; where the integers ®i ; °i are chosen in such a way that °i ¡ ®i C 1 D jKi j; j®i C °i j · 1 .i D 1; : : : ; s/; j.®1 C °1/ C ¢ ¢ ¢ C .®s C °s /j · 1: Proof. For any abelian group G and for any L1; : : : ; Ls µ G, x 2 G, denote by Nx .L1; : : : ; Ls / the number of solutions of .3:3/ .3:4/ and let a1 C ¢ ¢ ¢ C as D x I ai 2 Li ; N .L1; : : : ; Ls / D mx2aGx Nx .L1; : : : ; Ls /: Define K D K1 [ ¢ ¢ ¢ [ Ks . Let ': K 0 ! K be an Fs -isomorphism of a set of integers K 0 µ Z onto K , and let Ki0 be the preimage of Ki in K 0 .i D 1; : : : ; s/. Then evidently an equality with ai0; as0Ci 2 Ki0 holds if and only if holds, and it follows that a10 C ¢ ¢ ¢ C as0 D as0C1 C ¢ ¢ ¢ C a20s '.a10/ C ¢ ¢ ¢ C '.as0 / D '.as0C1/ C ¢ ¢ ¢ C '.a20s / N .K1; : : : ; Ks / D N .K 10; : : : ; Ks0 /: By Theorem 3.2, the right-hand side can only increase if, for all i D 1; : : : ; s, we replace Ki0 by Ki00 D f®i ; ®i C 1; : : : ; °i g µ Z, where ®i ; °i satisfy (3.3) and (3.4): N .K 10; : : : ; Ks0 / · N .K 100; : : : ; Ks00/: Now, let K i be the images of Ki00 under the canonical homomorphism Z ! Z= pZ. The assertion of the theorem follows from the observation that for any integer x 2 f¡b p=2c; ¡b p=2c C 1; : : : ; b p=2cg with the corresponding residue x 2 Z= pZ, Nx .K 100; : : : ; Ks00/ D Nx .K 1; : : : ; K s /; N .K 100; : : : ; Ks00/ D N .K 1; : : : ; K s /: and thus Notice, that in the proof above we could not apply Theorem 2.1 not only because the sets Ki are distinct, but also (and mainly) because there are no restrictions on the cardinalities j2Ki j. However, Theorem 3.1 works perfectly in this situation. 4. Erdo˝s–Heilbronn Conjecture Let h ¸ 2 be an integer and let K be a subset of the set of elements of an abelian group. Denote by h b K the set of all sums of h distinct elements from K : h b K D fa1 C ¢ ¢ ¢ C ah : a1; : : : ; ah 2 K and ai 6D aj for 1 · i < j · hg: Let p be a prime. Erdo˝s and Heilbronn (see [9, p. 95]) conjectured that any K µ Z= pZ satisfies j2b K j ¸ minf2jK j ¡ 3; pg: Note that the inequality j2b K j ¸ 2jK j ¡ 3 trivially holds for K µ Z (and for finite subsets of torsion-free abelian groups). In general, we have Proposition 4.1 (Folklore). .4:1/ .4:2/ .4:3/ (a) For any positive integer h and any finite set K µ Z we have jh b K j ¸ hjK j ¡ h2 C 1: (b) If jK j ¸ maxfh C2; 5g, then equality in (4.2) holds if and only if K is an arithmetic progression. For a proof see [19, Theorems 1 and 2], [5, App. C], or [20, Theorem 1.10]. After a number of partial results, say, [ 12 ], [ 18 ], and [ 21 ] (see [ 2 ] for more references), the Erdo˝s–Heilbronn conjecture (4.1) was finally settled by Dias da Silva and Hamidoune [ 8 ]. Another proof was suggested in [ 1 ]. Actually, Dias da Silva and Hamidoune proved a more general inequality jh b K j ¸ minfhjK j ¡ h2 C 1; pg for arbitrary h ¸ 2 and K µ Z= pZ. Recently Alon et al. [ 2 ] obtained a fairly general additive theorem which contains inequality (4.3) as a particular case. However, to the best of our knowledge, the problem of when the equality in (4.3) holds is still open. Here we obtain an answer for sufficiently small K µ Z= pZ as a direct consequence of Proposition 4.1(b) and Theorem 2.1. Theorem 4.1. For any h ¸ 2 there exists a constant c4 D c4.h/ with the following property. For any prime p and any set of residues K µ Z= pZ such that .4:4/ .4:5/ the equality maxfh C 2; 5g · jK j · c4 p; jh b K j D hjK j ¡ h2 C 1 holds if and only if K is an arithmetic progression. Proof. Put c4.h/ D minfh¡1; c1.2h; h/g, where c1 is defined in Theorem 2.1. If K µ Z= pZ is an arithmetic progression and jK j · p= h, then (4.5) obviously holds. Conversely, assume that K µ Z= pZ satisfies (4.4) and (4.5). Fix an .h ¡ 2/-element subset H µ K and denote L D K nH . Then j2b Lj · jh b K j < hjK j. Therefore jK C K j · j2b K jCjK j · j2b LjCjH C K jCjK j < hjK jC.h ¡2/jK jCjK j < 2hjK j: By Theorem 2.1, the set K is Fh -isomorphic to a set of integers K 0 µ Z. Then clearly jh b K 0j D jh b K j D hjK 0j ¡ h2 C 1, and by Proposition 4.1(b), K 0 is an arithmetic progression. Then so is K . Freiman et al. [ 12 ] applied a similar “rectification” approach for h D 2. Their technique is quite different and is not based on Theorem 2.2, and for h D 2 their result is much stronger than Theorem 4.1 above. However, the method of [ 12 ] does not extend to h ¸ 3. See also Ro¨dseth [ 21 ]. When the Erdo˝s–Heilbronn conjecture was proved, it had been conjectured by the second author that in fact a much more general result holds. Specifically, let K and L be subsets of an abelian group, such that jK j · jLj, and let ¿ : K ! L be an arbitrary ¿ mapping from K to L. Define K C L to be the set of all the sums a C b (where a 2 K ; b 2 L) such that b 6D ¿ .a/: ¿ K C L D fa C b: a 2 K ; b 2 L ; and b 6D ¿ .a/g: Conjecture 4.1 (Lev). Let K and L be subsets of Z= pZ satisfying jK j · jLj, and let ¿ : K ! L be an arbitrary mapping from K to L. Then ¿ jK C Lj ¸ minfjK j C jLj ¡ 3; pg: Using Theorems 2.1 or 3.1 we are able to prove this for small K ; L. First, we need a corresponding result in Z. Theorem 4.2. Let K and L be finite subsets of Z satisfying jK j · jLj, and let ¿ : K ! L be an arbitrary mapping from K to L. Then ¿ jK C Lj ¸ jK j C jLj ¡ 3: Proof. Write down the elements of K and L in ascending order: K D fa1; : : : ; ak g and L D fb1; : : : ; bl g, where ai < aj and bi < bj for i < j . We first assume that jK j < jLj. Then there exists bj 2 L which is not an image of an element of K under ¿ . Therefore among the k C l ¡ 1 distinct sums at most one sum in the first row and at most one sum in the last row are excluded by the ¿ condition b 6D ¿ .a/. At least k C l ¡ 3 remaining sums fall into K C L. Now assume jK j D jLj. Then either there exists bj 2 L which has no preimage in K , and we can repeat the argument above; or ¿ is a bijection, in which case we consider k C l ¡ 1 distinct sums a1 C b1 < a1 C b2 < ¢ ¢ ¢ < a1 C bl < a2 C bl < a3 C bl < ¢ ¢ ¢ < ak C bl ; ¿ and observe again, that at most two of them may not fall into K C L. Theorem 4.3. The assertion of Conjecture 4.1 holds provided that either L D K and jK j D jLj · c5 p (with a sufficiently small absolute constant c5), or jK j C jLj · log4 p. Proof. In the first case (L D K ; jK j D jLj · c5 p) we observe that ¿ jK C K j · jK C K j C jK j · 3jK j ¡ 4; ¿ assuming jK C K j < 2jK j ¡ 3. Then by Theorem 2.1, K is F2-isomorphic to a set of integers K 0. Let ¿ 0: K 0 ! K 0 be the mapping induced by ¿ . Then K 0 satisfies ¿0 jK 0 C K 0j < 2jK 0j ¡ 3, which, as Theorem 4.2 shows, is impossible for K 0 µ Z. In the second case (jK j C jLj · log4 p), we find, as in Theorem 3.3, a set of integers M µ Z which is F2-isomorphic to the union K [ L, define K 0; L0 µ M to be the preimages of K ; L ; respectively, and define ¿ 0: K 0 ! L0 to be the mapping induced by ¿ . Then by Theorem 4.2, ¿ ¿0 jK C Lj D jK 0 C L0j ¸ jK 0j C jL0j ¡ 3 D jK j C jLj ¡ 3: Using [22, Lemma 3.3] the last theorem can be extended to all sets K and L such that "jLj · jK j · jLj · c6."/ p for any " > 0. As a concluding remark, we note that the rectification method can be used not only for the group Z= pZ: for instance, in [ 4 ] it is applied for the torus Rm =Zm . Acknowledgment We would like to thank Dani Berend and the referee for pointing out some inaccuracies in the manuscript. 1. N. Alon , M. B. Nathanson , and I. Z. Ruzsa , Adding distinct congruence classes modulo a prime , Amer. Math. 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Y. F. Bilu, V. F. Lev, I. Z. Ruzsa. Rectification Principles in Additive Number Theory, Discrete & Computational Geometry, 1998, 343-353, DOI: 10.1007/PL00009351