Journal of Number Theory 104 (2004) 162 169 http://www.elsevier.com/locate/jnt Progression-free sets in finite abelian groups Vsevolod F. Lev Department of Mathematics, University of Haifa at Oranim, Tivon 36006, Israel Received 11 November 2002; revised 23 May 2003 Communicated by D. Goss Abstract Let G be a finite abelian group. Write 2G :¼ f2g: g and denote by rkð2gþ the rank of the group2g: Extending a result of Meshulam, we prove the following. Suppose that ADG is free of true arithmetic progressions; that is, a 1 þ a 3 ¼ 2a 2 with a 1 ; a 2 ; a 3 AA implies that a 1 ¼ a 3 : Then jajo2jgj=rkð2gþ: When G is of odd order this reduces to the original result of Meshulam. As a corollary, we generalize a result of Alon and show that if an integer k2 and a real e40 are fixed, j2gj is large enough, and a subset ADG satisfies jajð1=k þ eþjgj; then there exists A 0 DA such that 1pjA 0 jpk and the elements of A 0 add upto zero. When G is of odd order or cyclic this reduces to the original result of Alon. r 2003 Elsevier Inc. All rights reserved. 1. Motivation and background Since the fundamental paper of Roth [R53] one of the central problems in combinatorial number theory is to estimate the maximum possible cardinality of an integer set AD½1; mš; free of three-term arithmetic progressions; here m is a growing to infinity integer parameter. Equivalently, one can consider progression-free subsets of Z m ; the cyclic groupof order m: If DðZ m Þ denotes the maximum possible cardinality of such a subset, then the result of Roth can be formulated as DðZ m Þ¼ Oðm=ln ln mþ; and further refinements due to Heath-Brown [H87], Szemere di [S90], and Bourgain [B99] as DðZ m Þ¼Oðm=ln a mþ with an absolute constant a40: A natural generalization is to replace Z m with an arbitrary finite abelian group G and estimate DðGÞ; the maximum possible cardinality of a subset ADG; free of E-mail address: seva@math.haifa.ac.il. 0022-314/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/s0022-314(03)00148-3
V.F. Lev / Journal of Number Theory 104 (2004) 162 169 163 three-term arithmetic progressions. For abelian groups G of odd order, Brown and Buhler in [BB82] and independently Frankl et al. in [FGR87] proved that DðGÞ ¼ oðjgjþ as jgj-n: A quantitative estimate is due to Meshulam [M95] who has shown that DðGÞp2jGj=rkðGÞ for any non-trivial abelian group G of odd order; here and below throughout the paper rkðgþ denotes the rank of G: As Meshulam shows, along with the result of Heath-Brown and Szemere di this implies that DðGÞ ¼ OðjGj=ln b jgjþ for some absolute constant b40 and any non-trivial abelian group G of odd order. Suppose now that G is either abelian of odd order or finite cyclic. Using the result of Frankl et al. in the first case, and the result of Roth is the second case, Alon proved in [A87] the following. Suppose that an integer k2 and a real e40 are fixed. Then provided that jgj is large enough, for any subset ADG satisfying jaj4ð1=k þ eþjgj there exists A 0 DA such that 1pjA 0 jpk and the elements of A 0 add upto zero. 2. Notation and summary of results Our goal is to extend the results of Meshulam and Alon onto arbitrary finite abelian groups, not necessarily of odd order. First we discuss a question which we have not addressed yet, namely: what is a three-term arithmetic progression and when we say that a set is free of such progressions? The answer is quite straightforward for groups of odd order but somewhat subtler for even order groups: is the set f1; 6gDZ 10 free of arithmetic progressions? Let G be an abelian group. We say that a three-element sequence P ¼ða 1 ; a 2 ; a 3 Þ of elements of G is an arithmetic progression if a 1 þ a 3 ¼ 2a 2 ; and not all elements of P are equal to each other. If, moreover, all elements of P are pairwise distinct (equivalently, if a 1 aa 3 ) then we say that P is a true arithmetic progression. Clearly, if G has no elements of even order then any arithmetic progression is true and the notions of arithmetic progression and true arithmetic progression coincide. By PF[G] we denote the family of all subsets of G; free of true arithmetic progressions; that is, all subsets ADG such that a 1 þ a 3 ¼ 2a 2 with a 1 ; a 2 ; a 3 AA implies that a 1 ¼ a 3 : Assuming that G is finite, we let DðGÞ :¼ maxfjaj: AAPF½GŠg; the maximum possible cardinality of a subset of G; free of true arithmetic progressions. One last bit of notation before we can formulate our results. For an abelian group G we write 2G :¼ f2g: g: Clearly, 2G is a subgroupof G; and if G is finite of odd order then 2G ¼ G: Furthermore, 2G is the trivial groupif and only if G is an elementary abelian 2-group.
164 V.F. Lev / Journal of Number Theory 104 (2004) 162 169 Theorem 1. For any finite abelian group G such that 2G is non-trivial we have DðGÞo 2jGj rkð2gþ : Corollary 1. There exist absolute real constants C and b40 with the property that for any finite abelian group G such that 2G is non-trivial we have jgj DðGÞoC ðln j2gjþ b: When G is of odd order Theorem 1 and Corollary 1 reduce to [M95, Theorem 1.2] and [M95, Corollary 1.3], respectively. We prove Theorem 1 in the next section using a version of Meshulam s argument (which itself originates from Roth s method) and employing some additional ideas. Corollary 1 can be derived from Theorem 1 and the result of Heath-Brown and Szemere di in the same way as [M95, Corollary 1.3] is derived from [M95, Theorem 1.2]; the only difference is that instead of a large cyclic subgroup HpG one should consider a large cyclic subgroup Hp2G: We refer the reader to [M95] for the details. Theorem 2. Let G be a finite abelian group. Then for any integer k2 and any subset ADG satisfying jajð1=kþjgjþ4kdðgþ there exists A 0 DA such that 1pjA 0 jpk and the elements of A 0 add up to zero. Corollary 2. For any integer k2 and real e40 there exists an integer n 0 ðk; eþ with the following property. Let G be a finite abelian group such that j2gj4n 0 ðk; eþ and suppose that a subset ADG satisfies jajð1=k þ eþjgj: Then there exists A 0 DA such that 1pjA 0 jpk and the elements of A 0 add up to zero. When G is a finite cyclic group, Corollary 2 reduces to [A87, Theorem 1.1] and Theorem 2 is analogous to [A87, Proposition 2.5]. The proof of Theorem 2 can be conducted following the proof of [A87, Proposition 2.5], except that the threeuniform hypergraph considered by Alon is to be replaced by the graph in which two vertices are adjacent if and only if the corresponding elements are the endterms of a true three-term arithmetic progression. For completeness and reader s convenience we outline the proof at the end of the next section; details can be retrieved from [A87]. Corollary 2 is immediate from Theorem 2 and Corollary 1. 3. Proof of Theorems 1 and 2 Proof of Theorem 1. Using induction on the integer n1; we show that if G is a finite abelian groupsuch that rkð2gþn; then DðGÞo2jGj=n: The case n ¼ 1 is trivial and we assume that n2: Suppose that AAPF½GŠ; we want to show that jajo2jgj=n:
V.F. Lev / Journal of Number Theory 104 (2004) 162 169 165 Consider the automorphism j A AutðGÞ defined by jðgþ¼2gðþ: The image of j is 2G and the kernel of j is G 0 :¼f: 2g ¼ 0g: Thus G=G 0 D2G and writing s :¼ jg 0 j and t :¼ j2gj we have st ¼jGj: The set A is a union of t disjoint subsets contained in a G 0 -coset each. Let n 1 ; y; n t denote the cardinalities of these subsets so that n 1 þ? þ n t ¼jAj and 0pn i pjg 0 ði ¼ 1; y; tþ: By  we denote the Fourier coefficients of the characteristic function of A; that is, for a character waĝ we write ÂðwÞ :¼ aaa %wðaþ; where %w is the character conjugate to w: The cartesian square and cartesian cube of A are denoted by A 2 and A 3 ; respectively. Since AAPF½GŠ using orthogonality relations we obtain ðâðwþþ2 Âð%w 2 Þ¼jGj#fða 1 ; a 2 ; a 3 ÞAA 3 : a 1 þ a 3 ¼ 2a 2 g waĝ ¼jGj#fða 1 ; a 2 ÞAA 2 :2a 1 ¼ 2a 2 g ¼jGj#fða 1 ; a 2 ÞAA 2 : a 2 a 1 AG 0 g ¼jGjðn 2 1 þ? þ n2 t Þ p jgjðn 1 þ? þ n t Þ max 1pipt n i p jgjjajjg 0 j: We single out those summands corresponding to real characters w: Notice that w is real if and only if w 2 ¼ w 0 ; the principle character. In this case Âð%w2 Þ¼jAj and we get waĝ: w2 aw 0 ðâðwþþ2 Âð%w 2 Þ ðâðwþþ2 Âð%w 2 Þ jgjjajjg 0 j waĝ: w2 ¼w 0 ¼jAj ðâðwþþ2 jgjjajjg 0 j waĝ: w2 ¼w 0 ¼jAjS jgjjajjg 0 j; ð1þ where S ¼ P waĝ: w2 ¼w 0 ðâðwþþ2 : Set M :¼ max jâð%w2 Þj: waĝ: w2 aw 0
166 V.F. Lev / Journal of Number Theory 104 (2004) 162 169 Since ÂðwÞ is real for any real character w; from (1) and the Parseval identity it follows that jajs jgjjajjg 0 jp M jâðwþj2 waĝ: w2 aw 0 0 1 ¼ M@ jâðwþj2 SA waĝ ¼ðjAjjGj SÞM: ð2þ Fix now a non-real character w such that jâð%w2 Þj ¼ M and let WoG be the kernel of w 2 ; that is, W ¼f: w 2 ðgþ ¼1g: We have Âð%w 2 Þ¼ w 2 ðaþ aaa 0 ¼ 1 @ w 2 ðgþ jwj ¼ 1 jwj ¼ 1 jwj aaa-ðgþwþ 1 1A w 2 ðgþjða gþ-wj ð w 2 ðgþþðdðwþ jða gþ-wjþ: As ðða gþ-wþapf½wš for any ; all differences DðWÞ jða gþ-wj are non-negative and writing for brevity k :¼ DðWÞ=jWj we obtain M ¼jÂð%w2 Þjp 1 jwj ðdðwþ jða gþ-wjþ ¼ kjgj jaj: Comparing with (2) we conclude that after simplifications, jajs jgjjajjg 0 jpðjajjgj SÞðkjGj jajþ; ks þjaj 2 pjajjg 0 jþkjajjgj: ð3þ To estimate S we represent A as a union of s disjoint subsets contained in a 2G-coset each. (Recall that ½G : 2GŠ ¼jG 0 j¼s:) We denote the cardinalities of these subsets by m 1 ; y; m s so that m 1 þ? þ m s ¼jAj and 0pm i pj2gjði ¼ 1; y; sþ: Using
orthogonality relations once again we get S ¼! 1 w 2 ðgþ jgj jâðwþj2 waĝ ¼ 1 w 2 jgj ðgþjâðwþj2 waĝ ¼ #fða 1 ; a 2 ÞAA 2 : a 1 a 2 ¼ 2gg ¼jG 0 j#fða 1 ; a 2 ÞAA 2 : a 2 a 1 A2Gg ¼jG 0 jðm 2 1 þ? þ m2 s Þ jg 0 jðm 1 þ? þ m s Þ 2 =s ¼jAj 2 : Combined with (3) this yields ARTICLE IN PRESS V.F. Lev / Journal of Number Theory 104 (2004) 162 169 167 jaj jg 0 jpkðjgj jajþ: ð4þ We now use the induction hypothesis as applied to W to estimate k ¼ DðWÞ=jWj: Since the quotient group G=W is isomorphic to the image of w 2 ; which is a finite subgroup of the multiplicative group of the field of complex numbers, we conclude that G=W is cyclic or trivial. It follows that 2G=2W is cyclic or trivial, too, whence rkð2wþrkð2gþ 1n 1 and therefore ko2=ðn 1Þ: Substituting to (4) we get which gives jaj jg 0 jo2ðjgj jajþ=ðn 1Þ jajo 2=ðn 1ÞþjG 0j=jGj jgj: 2=ðn 1Þþ1 ð5þ Since the quotient jg 0 j=jgj can be rather large, at this point we have to take into account the algebraic structure of G: Write r :¼ rkðgþ and G ¼ " r i¼1 G i; where the direct summands are cyclic subgroups of G the orders of which d i :¼jG i jði ¼ 1; y; rþ satisfy 2pd 1 j?jd r : Set j :¼ maxfia½1; nš: d i ¼ 2g (with the understanding that max Ø ¼ 0) and put G 0 :¼ Mj i¼1 G i ; G 00 :¼ Mr i¼jþ1 G i :
168 V.F. Lev / Journal of Number Theory 104 (2004) 162 169 Consequently, we have G ¼ G 0 "G 00 ; 2G 0 ¼f0g; and rkðg 00 Þ¼rkð2G 00 Þ¼rkð2GÞn: ð6þ We distinguish two cases. First we assume that G satisfies rkð2gþ ¼rkðGÞ: In this case G 0 is the trivial groupand jg 0 j jgj ¼ Y 1 Y 2 p 1 r p 1 2 d i 2 2 no nðn 1Þ : ð7þ From (5) and (7) we obtain d ia½1;rš: d i is odd i ia½1;rš: d i is even 2=ðn 1Þþ2=nðn 1Þ jajo jgj ¼ 2 2=ðn 1Þþ1 n jgj; as required. It remains to consider the case when rkð2gþorkðgþ; that is, the group G 0 is nontrivial. To this end we observe that DðGÞpjG 0 jdðg 00 Þ: Indeed, if BDG satisfies jbj4jg 0 jdðg 00 Þ; then there exists a coset g þ G 00 (with some ) such that jb-ðg þ G 00 Þj4DðG 00 Þ; by the definition of DðG 00 Þ; the set B gdg 00 contains a true arithmetic progression, and so does B: Now by (6) and the argument above, as applied to the group G 00 ; we have DðGÞpjG 0 jdðg 00 ÞojG 0 j 2jG00 j n This completes the proof of Theorem 1. & ¼ 2jGj n : Proof of Theorem 2 (Sketch). Suppose that G; k; and A are as in the theorem. We can assume that k3 as otherwise the assertion follows readily from the pigeonhole principle. Define A 0 to be the set of all those elements of A which are the midterms of at least 2ðk 1Þ true three-term arithmetic progressions with the elements in A: A 0 :¼faAA: #fða 1 ; a 2 ÞAA 2 : a 1 þ a 2 ¼ 2a; a 1 aa 2 g2ðk 1Þg: Write A 00 :¼ A\A 0 and consider the graph G ¼ðA 00 ; EÞ on the vertex set A 00 in which two (distinct) vertices a 1 ; a 2 AA 00 are adjacent if and only if there exists aaa 00 such that a 1 þ a 2 ¼ 2a: The number of edges of G is jejpðk 2ÞjA 00 j (as any aaa 00 satisfies at most 2ðk 2Þ equalities of the form a 1 þ a 2 ¼ 2a with a 1 ; a 2 AA 00 ; a 1 aa 2 ), hence by a simple probabilistic argument there is an independent vertex set BDA 00 of cardinality jbjja 00 j 2 =ð4jejþja 00 j=ð4ðk 2ÞÞ: By the construction we have BAPF½GŠ whence jbjpdðgþ; therefore ja 00 jp4ðk 2ÞDðGÞ
V.F. Lev / Journal of Number Theory 104 (2004) 162 169 169 and consequently ja 0 j4jgj=k: By [A87, Corollary 2.3] there exists an integer ha½1; kš and a sequence ða 1 ; y; a h Þ of not necessarily distinct elements of A 0 such that a 1 þ? þ a h ¼ 0: By the choice of the set A 0 ; for any ia½1; hš there are at least 2ðk 1Þ representations 2a i ¼ a ð1þ i þ a ð2þ i with a ð1þ i ; a ð2þ i AA and a ð1þ i aa ð2þ i ; and it is not difficult to conclude that the sum a 1 þ? þ a h can be re-written as a sum of h pairwise distinct elements of A; see the proof of [A87, Proposition 2.5] for the explanation. Thus we have found ha½1; kš pairwise distinct elements of A with zero sum. & References [A87] N. Alon, Subset sums, J. Number Theory 27 (1987) 196 205. [B99] J. Bourgain, On triples in arithmetic progression, Geom. Funct. Anal. 9 (1999) 968 984. [BB82] T.C. Brown, J.C. Buhler, A density version of a geometric Ramsey theorem, J. Combin. Theory Ser. A 32 (1982) 20 34. [FGR87] P. Frankl, G. Graham, V. Ro dl, On subsets of abelian groups with no 3-term arithmetic progression, J. Combin. Theory Ser. A 45 (1987) 157 161. [H87] R. Heath-Brown, Integer sets containing no arithmetic progressions, J. London Math. Soc. 35 (1987) 385 394. [M95] R. Meshulam, On subsets of finite abelian groups with no 3-term arithmetic progressions, J. Combin. Theory Ser. A 71 (1995) 168 172. [R53] K.F. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953) 104 109. [S90] E. Szemerédi, Integer sets containing no arithmetic progressions, Acta Math. Hungar. 56 (1990) 155 158.