# #A1 INTEGERS 14 (2014) UPPER AND LOWER BOUNDS ON B

## Full text

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UPPER AND LOWER BOUNDS ON Bk+-SETS

Craig Timmons1

Dept. of Mathematics, University of California San Diego, La Jolla, California ctimmons@ucsd.edu

Received: 11/11/12, Revised: 10/2/13, Accepted: 12/6/13, Published: 1/6/14

Abstract

LetG be an abelian group. A setA⇢Gis a Bk+-set if whenevera1+· · ·+ak = b1+· · ·+bk withai, bj 2A, there is aniand ajsuch thatai=bj. IfAis aBk-set then it is also a Bk+-set, but the converse is not true in general. Determining the largest size of aBk-set in the interval {1,2, . . . , N}⇢Zor in the cyclic group ZN

is a well-studied problem. In this paper we investigate the corresponding problem for Bk+-sets. We prove nontrivial upper bounds on the maximum size of aBk+-set contained in the interval {1,2, . . . , N}. For odd k 3, we constructBk+-sets that have more elements than theBk-sets constructed by Bose and Chowla. We prove that any B+3-set A ⇢ ZN has at most (1 +o(1))(8N)1/3 elements. A set A is a Bk-set if whenevera1+· · ·+ak =ak+1+· · ·+a2k withai2A, there is ani6=j such thatai =aj. We obtain new upper bounds on the maximum size of aBk-set A⇢{1,2, . . . , N}, a problem first investigated by Ruzsa.

1. Introduction

LetGbe an abelian group. A setA⇢Gis aBk+-set if

a1+· · ·+ak =b1+· · ·+bk with a1, . . . , ak, b1, . . . , bk2A (1) impliesai=bj for some iandj. A setAis aBk-set if (1) implies (a1, . . . , ak) is a permutation of (b1, . . . , bk). IfAis aBk-set thenAis also aBk+-set, but in general the converse is not true. OftenB2-sets are calledSidon sets and have received much attention since they were first studied by Erd˝os and Tur´an  in 1941. LetFk(N) be the maximum size of a Bk-setA⇢[N] and letCk(N) be the maximum size of aBk-setA⇢ZN. IfA⇢ZN is aBk-set, thenAis also aBk-set when viewed as a subset ofZ. Thus, for anyk 2,Ck(N)Fk(N).

Erd˝os and Tur´an provedF2(N)N1/2+O(N1/4). Their argument was used by Lindstr¨om  to showF2(N)N1/2+N1/4+ 1. In 2010, Cilleruelo  obtained

1The author was partially supported by NSF Grant DMS-1101489 through Jacques Verstra¨ete.

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F2(N)N1/2+N1/4+12 as a consequence of a more general result. This is the best known upper bound onF2(N). By counting di↵erencesa b witha6=b, it is easy to prove C2(N)p

N+ 1. There are several constructions of dense B2-sets (see , , ) that showC2(N) N1/2for infinitely manyN. These constructions implyF2(N)⇠p

N and lim supCp2(N)N = 1.

Fork 3, bounds onFk(N) andCk(N) are not as precise. For eachk 2 and prime power q, Bose and Chowla  constructed aBk-setA⇢Zqk 1 with|A|=q so that

(1 +o(1))N1/kFk(N).

The current upper bounds onFk(N) andCk(N) do not match this lower bound for anyk 3. IfA⇢[N] is aBk-set then eachk-multiset inAgives rise to a unique sum in {1, . . . , kN}. Therefore, |A|+kk 1  kN which impliesFk(N) (k!·kN)1/k. Similar counting shows Ck(N)  (k!N)1/k. By considering di↵erences one can improve these bounds. We illustrate this idea with an example that is relevant to our results. Let A ⇢ZN be a B3-set. There are |A|2 (|A| 2) sums of the form a1+a2 a3 where a1, a2, anda3 are distinct elements ofA. Let A(2) ={{x, y}: x, y 2 A, x 6= y}. It is not hard to check that each n 2 ZN has at most one representation as n=a1+a2 a3 with{a1, a2}2A(2) anda32A\{a1, a2}. This implies |A|2 (|A| 2)N so|A|(2N)1/3+ 2. In general, for anyk 2

Ck(N)

✓ k 2

!

⇠k 2

!N

1/k

+Ok(1), (2)

and

Fk(N)

✓ k 2

!

⇠k 2

⇠k 2

⇡ N

1/k

+Ok(N1/2k). (3) These bounds were first obtained by Jia  in the even case, and Chen  in the odd case. The best upper bounds on Fk(N) are to due to Green . For every k 2, (3) has been improved (see for example  or ), but there is no value ofk 3 for which (2) has been improved. This is interesting since all of the constructions take place in cyclic groups and provide lower bounds onCk(N). For other bounds on Bk-sets the interested reader is referred to Green , Cilleruelo , O’Bryant’s survey , or the book of Halberstam and Roth .

Now we discuss Bk+-sets. WriteFk+(N) for the maximum size of aBk+-setA ⇢ [N], and Ck+(N) for the maximum size of a B+k-set A ⇢ ZN. Ruzsa  proved that a set A⇢[N] with no solution to the equation x1+· · ·+xk =y1+· · ·+yk in 2k distinct integers has at most (1 +o(1))k2 1/kN1/k elements. Call such a set a Bk-set and define Fk(N) in the obvious way. Any Bk+-set is also a Bk-set so thatFk+(N)Fk(N). Using the constructions of Bose and Chowla  and Ruzsa’s Theorem 5.1 of , we get for everyk 3,

(1 +o(1))N1/kFk(N)Fk+(N)Fk(N)(1 +o(1))k2 1/kN1/k.

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In this paper we improve this upper bound on Fk+(N) and Fk(N). We also improve this lower bound onFk+(N) for all odd k 3, and we prove a nontrivial upper bound on C3+(N). We do not consider the case when k = 2. The reason for this is that Ruzsa  provedF2(N)N1/2+ 4N1/4+ 11, and thusF2(N)⇠ F2+(N)⇠F2(N)⇠N1/2. In fact, aB2-set is the same as aB2+-set.

Our first result is a construction which shows that for any oddk 3, there is a B+k-set in [N] that has more elements than any knownBk-set contained in [N].

Theorem 1.1. For any prime power q and odd integer k 3, there is a Bk+-set A⇢Z2(qk 1)with |A|= 2q.

Using known results on densities of primes (see  for example), Theorem 1.1 implies

Corollary 1.2. For any integerN 1and any odd integer k 3, Fk+(N) (1 +o(1))21 1/kN1/k.

Green provedF3(N)(1 +o(1))(3.5N)1/3. We will use a Bose-ChowlaB3-set to construct a B3+-setA⇢[2q3] with|A|= 2q= (4·2q3)1/3. Putting the two results together we see thatAis denser than anyB3-set in [2q3] for sufficiently large prime powersq. Our construction and Green’s upper bound show thatF3(N) andF3(N) are not asymptotically the same.

The proof of Theorem 1.1 is based on a simple lemma, Lemma 2.1, which implies 2Ck(N)Ck+(2N) for any odd k 3. (4) This inequality provides us with a method of estimating Ck(N) by proving upper bounds on Ck+(N) for odd k. Our next theorem provides such an estimate when k= 3.

Theorem 1.3. If A⇢ZN is aB3+-set, then|A|(1 +o(1))(8N)1/3. Theorem 1.3 and (4) imply

Corollary 1.4. If A⇢ZN is aB3-set, then|A|(1 +o(1))(2N)1/3.

As shown above, there is a simpler argument that implies this bound. The novelty here is that our results imply (2) fork= 3. It is important to mention that the error term we obtain is larger than the error term in the bound C3(N)(2N)1/3+ 2.

We feel that any improvement in the leading term of Theorem 1.3 or (2) would be significant.

InZwe obtain the following bounds for smallk.

Theorem 1.5. (i) IfA⇢[N] is aB3+-set, then|A|(1 +o(1))(18N)1/3. (ii) If A⇢[N]is aB4+-set, then |A|(1 +o(1))(272N)1/4.

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Recall that Ruzsa  proved Fk(N)  (1 +o(1))k2 1/kN1/k which implies Fk+(N)  (1 +o(1))k2 1/kN1/k. For k 5, we were able to improve this upper bound on Fk+(N) by modifying arguments of Ruzsa. Our method also applies to Bk-sets. As a consequence, we improve the upper bound on Fk(N) for allk 3.

We state our result only for k= 3 and for largek. For other small values ofkthe reader is referred to Table 1 in Section 6.

Theorem 1.6. IfA⇢[N]is a B3-set, then|A|(1 +o(1))(162N)1/3. IfA⇢[N] is aBk-set, then

|A|

✓1 4+✏(k)

We remark that Ruzsa’s upper bound onFk(N) is asymptotic tok2N1/k. Our results do not rule out the possibility ofFk+(N) being asymptotic toFk(N).

Problem 1.7. Determine whether or notFk+(N) is asymptotic toFk(N) fork 3.

IfA⇢[N] is aBk-set, then the number of solutions to 2x1+x2+· · ·+xk 1= y1+· · ·+yk withxi, yj2Aiso(|A|k) (see ). ABk-set allows solutions to this equation withx1, . . . , xk 1, y1, . . . , yk all distinct, but such a solution cannot occur in a Bk+-set. If it were true that Fk+(N) is asymptotic toFk(N), then this would confirm the belief that it is the sums ofkdistinct elements ofAthat control the size ofAand the lower order sums should not matter. Jia  defines asemi-Bk-set to be a setAwith the property that all sums consisting ofkdistinct elements ofAare distinct. He states that Erd˝os conjectured  that a semi-Bk-set A ⇢[N] should satisfy|A|(1 +o(1))N1/k. A positive answer to Problem 1.7 would be evidence in favor of this conjecture.

At this time we do not know how to constructB+2k-sets orB2k-sets for anyk 2 that are bigger than the corresponding Bose-Chowla B2k-sets. We were able to construct interestingB4+-sets in the non-abelian setting.

LetGbe a non-abelian group. A setA⇢Gis anon-abelian Bk-set if

a1a2· · ·ak=b1b2· · ·bk with ai, bj 2A (5) impliesai=bi for 1ik. IfA⇢Gis a non-abelianBk-set, then everyk-letter word in |A| is di↵erent so |A|k |G|. Odlyzko and Smith  proved that there exist infinitely many groups Gsuch thatG has a non-abelian B4-set A⇢Gwith

|A|= (1 +o(1))⇣

|G|

1024

1/4

. They actually proved a more general result that gives constructions of non-abelian Bk-sets for all k 2. The case when k = 4 is the only result that we will need. Define a non-abelian B+k-set to be a set A ⇢ G such that (5) impliesai =bi for some i2{1,2, . . . , k}. As in the abelian setting, a non-abelian Bk-set is also a non-abelian B+k-set but the converse is not true in general. Using a construction of , we prove

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Theorem 1.8. For any prime p with p 1 divisible by 4, there is a non-abelian groupGof order48(p4 1) that contains a non-abelianB4+-set A⇢Gwith

|A|= 1 2(p 1).

Our result shows that there are infinitely many groupsGsuch thatGhas a non- abelian B4+-set Awith |A| =⇣

|G|

768

1/4

+o(|G|1/4). We conclude our introduction with the following conjecture concerningB2k+-sets.

Conjecture 1.9. Ifk 4 is any even integer, then there exists a positive constant ck such that for infinitely manyN,

Fk+(N) (1 +ck+o(1))N1/k.

If Conjecture 1.9 is true with ck = 21 1/k 1 as in the odd case, then using Green’s upper boundF4(N)(1 +o(1))(7N)1/4, we can conclude thatF4(N) and F4(N) are not asymptotically the same just as in the case when k= 3. Our hope is that a positive answer to Conjecture 1.9 will either provide an analogue of (4) for evenk 4, or a construction of aBk+-set that does not use Bose-ChowlaBk-sets.

2. Proof of Theorem 1.1

In this section we show how to constructBk+-sets for oddk 3. Our idea is to take a denseBk-setA and a translate ofA.

Lemma 2.1. If A⇢ZN is aBk-set wherek 3is odd, then A+:={a+bN :a2A, b2{0,1}}

is aB+k-set inZ2N.

Proof. Letk 3 be odd and suppose Xk

i=1

ai+biN ⌘ Xk i=1

ci+diN (mod 2N) (6)

whereai, ci 2A, andbi, di2{0,1}. Taking (6) moduloN gives Xk

i=1

ai⌘ Xk

i=1

ci (mod N).

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Since A is aBk-set in ZN, (a1, . . . , ak) must be a permutation of (c1, . . . , ck). If we label theai’s andci’s so thata1a2· · ·ak and c1 c2 · · ·ck, then ai=ci for 1ik. Rewrite (6) as

Xk i=1

biN ⌘ Xk i=1

diN (mod 2N).

The sums Pk

i=1bi and Pk

i=1di have the same parity. Since k is odd and bi, di 2 {0,1}, there must be ajsuch thatbj =dj, soaj+bjN⌘cj+djN (mod 2N).

Let q be a prime power, k 3 be an odd integer, and Ak be a Bose-Chowla Bk-set withAk ⇢Zqk 1 (see  for a description ofAk). Let

A+k ={a+b(qk 1) :a2Ak, b2{0,1}}.

By Lemma 2.1, A+k is aB+k-set in Z2(qk 1) and |A+k| = 2|Ak| = 2q. This proves Theorem 1.1.

3. Proof of Theorem 1.3

LetA⇢ZNbe aB3+-set. IfNis odd, then 2x⌘2y(modN) impliesx⌘y(modN).

IfNis even, then 2x⌘2y(modN) impliesx⌘y(modN) orx⌘y+N/2 (modN).

Because of this, the odd case is quite a bit easier to deal with and so we present the more difficult case. In this sectionN is assumed to be even. IfN is odd, then the proof of Theorem 1.5(i) given in the next section works inZN. The only modification needed is to divide byN instead of 3Nwhen applying Cauchy-Schwarz.

For simplicity of notation, we writex=y rather thanx⌘y (modN).

Forn2ZN, define f(n) = #n

({a, c}, b)2A(2)⇥A:n=a b+c,{a, c}\{b}=;o . Recall thatA(2)={{x, y}:x, y2A, x6=y}. The sumP

f(n)(f(n) 1) counts the number of ordered pairs (({a, c}, b),({x, z}, y)) such that the tuples ({a, c}, b) and ({x, z}, y) are distinct, and both are counted byf(n). For each such pair we cannot have{a, c}={x, z}. Otherwise, the tuples would be equal. If (({a, c}, b),({x, z}, y)) is counted by P

f(n)(f(n) 1), thena+y+c=x+b+z. By the B3+ property, {a, y, c}\{x, b, z}6=;so that{a, c}\{x, z}6=;orb=y. The tuples are distinct, so both of these cases cannot occur at the same time.

Case 1: {a, c}\{x, z}6=;andb6=y.

Without loss of generality, assume a = x. Cancel a from both sides of the equation a b+c =x y+z and solve forc to getc =b y+z. Here we are

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using the ordering of the tuples (({a, c}, b),({x, z}, y)) to designate which element is solved for after the cancellation of the common term.

If z=b, thenc+y = 2b and we have a 3-term arithmetic progression (a.p. for short). The number of trivial 3-term a.p.’s inAis at most 2|A|since for anya2A,

a+a= 2a= 2(a+N/2).

Next we count the number of nontrivial 3-term a.p.’s. By nontrivial, we mean that all terms involved in the a.p. are distinct, anda+a= 2(a+N/2) is considered to be trivial.

Ifp+q= 2ris a 3-term a.p., then callpand qouter terms. Letpbe an outer term of the 3-term a.p. p+q = 2r where p, q, r 2 A. We will show that pis an outer term of at most one other nontrivial a.p. Let p+q0 = 2r0 be another a.p.

withq0, r0 2Aand (q, r,)6= (q0, r0).

Ifr=r0, thenp+q= 2r= 2r0 =p+q0 so q=q0. This is a contradiction and so we can assume thatr6=r0.

If q = q0, then 2r = p+q = p+q0 = 2r0 so r0 = r or r0 = r+N/2. Thus, p+q= 2rorp+q= 2(r+N/2).

Now supposer6=r0 andq6=q0. Since 2r q=p= 2r0 q0 we have by theB+3 property,

{r, q0}\{r0, q}6=;.

The only two possibilities arer=qorr0=q0, but in either of these cases we get a trivial 3-term a.p. Putting everything together proves the following lemma.

Lemma 3.1. If A ⇢ZN is a B+3-set, then the number of 3-term arithmetic pro- gressions inA is at most4|A|.

Given a fixed elementa2Aand a fixed 3-term a.p. c+y = 2b inA, there are at most 4! ways to form an ordered tuple of the form (({a, c}, b),({a, b}, y)). The number of ordered tuples counted by P

f(n)(f(n) 1) when {a, c}\{x, z} 6= ; andz =bis at most 4!|A| ·4|A|= 96|A|2. The first factor of|A| in the expression 4!|A| ·3|A|comes from the number of ways to choose the elementa.

Assume now that z6=b. Recall that we have solved for c to getc =b y+z.

Ifb=y, thenc=z which implies{a, c}={x, z}, a contradiction as the tuples are distinct. By definitiony6=z, soc=b y+zwhere{b, z}2A(2)and{y}\{b, z}=;. The number of ways to writec in this form isf(c). Given such a solution{b, z}, y counted byf(c), there are two ways to orderbandz, and|A|ways to choosea=x.

The number of ordered tuples we obtain when{a, c}\{x, z}6=;andz 6=b is at most|A| ·2P

c2Af(c). This completes the analysis in Case 1.

Before addressing Case 2, the case when b = y and {a, c}\{x, z} = ;, some additional notation is needed. Ford2A+A, define

S(d) = {a, b}2A(2) :a+b=dand there is a pair{a0, b0}2A(2) with{a, b}\{a0, b0}=;anda0+b0=d}.

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Letd1, d2, . . . , dM be the integers for whichS(di)6=;. WriteSi2forS(di) and define Ti1={a:a2{a, b}for some{a, b}2S2i}.

Letsi=|Si2|andd1, d2, . . . , dmbe the integers for whichsi= 2. Letdm+1, . . . , dM be the integers for which si 3. For 1 i M, we will use the notationSi2 = {{ai1, bi1},{ai2, bi2}, . . . ,{aisi, bisi}}. A simple, but important, observation is that for any fixed i2{1, . . . , M}, any element ofAappears in at most one pair in Si2.

If A was a B3-set, then there would be no di’s. This suggests that a B3+-set or a B3-set that is denser than a B3-set should have many di’s. The B3+-set A+3 constructed in Theorem 1.1 hasm⇡ 12 |A2+3| . However, ifA+3 is viewed as a subset ofZ, thenm⇡14 |A2+3| (see Lemma 4.3 which also holds inZN ifN is odd).

Case 2: b=y and{a, c}\{x, z}=;.

Ifb=y, thena+c=x+z. There are|A|choices forb=y and XM

i=1

|Si2|(|Si2| 1)

ways to choose an ordered pair of di↵erent sets{a, c},{x, z}2A(2)witha+c=x+z, and{a, c}\{x, z}=;.

Putting Cases 1 and 2 together gives the estimate Xf(n)(f(n) 1)|A| 2X

c2A

f(c) + XM i=1

|Si2|(|Si2| 1)

!

+ 96|A|2. (7) Our goal is to find upper bounds on the sumsP

c2Af(c) andPM

i=1|S2i|(|Si2| 1).

Lemma 3.2. If x2Ti1\Tj1 for some i6=j, then (i) max{si, sj}3and (ii) if si=sj = 3, then for somex1, y, z2Adepending oniandj, we havedj =di+N/2 andSi2={{x, x1},{y, z},{y+N2, z+N2}},Sj2={{x, x1+N2},{y+N2, z},{y, z+N2}}. Proof. If si = 2 and sj = 2 then we are done. Assume sj > 2. Let Si2 = {{ai1, bi1}, . . . ,{aisi, bisi}} and Sj2 ={{aj1, bj1}, . . . ,{ajsj, bjsj}}. Without loss of gen- erality, suppose x = a1i and x = a1j. By definition, si 2 so we can write di=x+bi1=ai2+bi2 anddj=x+bj1=aj2+bj2=aj3+bj3.

Solve forxto getx=ai2+bi2 bi1=aj2+bj2 bj1. This can be rewritten as ai2+bi2+bj1=aj2+bj2+bi1. (8) Sincedi6=dj,bi1 cannot bebj1 thereforebj1is not on the right hand side of (8), and bi1 is not on the left hand side of (8). By theB+3 property,{ai2, bi2}\{aj2, bj2}6=;.

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The same argument can be repeated withaj3in place ofaj2andbj3in place ofbj2 to get

{ai2, bi2}\{aj3, bj3}6=;.

Recall any element of Acan occur at most once in the list aj1, bj1, aj2, bj2, . . . ajsj, bjsj thussj 3. By symmetry,si3.

Now suppose si = sj = 3. Repeating the argument above, we have for each 2k3 and 2l3,

|{ail, bil}\{ajk, bjk}|= 1.

This intersection cannot have size 2 since di 6=dj. Without loss of generality, let y = ai2 = aj2, z = bi2 = aj3, u = ai3 = bj2, and v = bi3 = bj3. We represent these equalities betweenTi1 andTj1 using a bipartite graph with partsTi1 andTj1 where w2Ti1 is adjacent tow0 2Tj1 if and only ifw=w0 (see Figure 1).

s s s s s s

s s s s s s

@@

@@

@@

aj1=x ai1=x

bj1 bi1

aj2=y ai2=y

bj2=u bi2=z

aj3=z ai3=u

bj3=v bi3=v

j1

### T

i1

Figure 1 - Equality Graph for Lemma 3.2

The equalitiesdi=y+z=u+vanddj =y+u=z+vimplydi dj=z uand di dj =u z. Therefore 2z= 2u. Ifz=u, then this is a contradiction since the elements in the listx, bi1, y, z, u, v are all distinct. It is in this step that the parity ofN plays an important role. We concludeu=z+N/2 and

dj =y+u=y+ (z+N/2) =y+z+N/2 =di+N/2.

Letbi1=x1so bj1=x1+N/2. Sincedi=y+z=u+v andu=z+N/2, v=y+z u=y+z (z+N/2) =y N/2 =y+N/2.

Substitutingu=z+N/2 and v=y+N/2 gives the assertion about the pairs in Si2 andSj2 whensi=sj= 3.

Corollary 3.3. Ifsi 4, then for anyj6=i,Ti1\Tj1=;. Furthermore, anyx2A is in at most two Ti1’s with si= 3.

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Proof. The first statement follows immediately from Lemma 3.2. For the second statement, suppose x 2 Ti1\Tj1 with si = sj = 3 and i 6= j. By Lemma 3.2, {x, x1}2Si2 and{x, x1+N/2}2Sj2 for somex1 2A. Ifx2Tk1 withk6=i, then {x, x1+N/2}2S2k sodj =x+ (x1+N/2) =dk andj=k.

Lemma 3.4. If A⇢ZN is aB3+-set, then X

c2A

f(c)|A|2+ 7|A|. Proof. Forc2A, let

g1(c) = #n

({x, z}, y)2A(2)⇥A:c=x y+z, c6=y,{x, z}\{y}=;o and

g2(c) = #n

({x, z}, y)2A(2)⇥A:c=x y+z, c=y,{x, z}\{y}=;o . For each c2 A, f(c) =g1(c) +g2(c). The sumP

c2Ag2(c) is exactly the number of nontrivial 3-term a.p.’s in A. By Lemma 3.1, P

c2Ag1(c) 4|A|. Estimating P

c2Ag1(c) takes more work. To computeg1(c) with c 2A, we first choose ani withc2Ti1, and then choose one of the pairs{x, z}2Si2\{c, y}to obtain a solution c=x y+zwithc6=yand{x, z}\{y}=;.

Ifc /2T11[· · ·[TM1, then the equationc+y=x+zwithc, y, x, andzall distinct has no solutions inA sog1(c) = 0. Assumec2T11[· · ·[TM1.

Case 1: c /2T11[· · ·[Tm1.

By Corollary 3.3, there are two possibilities. One is that there is a uniquejwith c2Tj1and sj 3. In this case,|S2j| |A|2 so g1(c) |A|2 . The other possibility is thatc 2Ti1\Tj1 withsi =sj = 3 and i 6=j. In this case, g1(c)4 because we can choose eitheri or j, and then one of the two pairs in S2i or Sj2 that does not containc.

Case 2: c2T11[· · ·[Tm1.

By Lemma 3.2,cis not in anyTj1 withsj 4 andc is in at most twoTj1’s with sj = 3. There are at most |A| Ti1’s withc 2Ti1 since there are at most|A| pairs {c, y}that containc sog1(c)|A|+ 4.

In all cases,g1(c)|A|+ 4 and X

c2A

f(c) =X

c2A

(g1(c) +g2(c))|A|(|A|+ 4) + 4|A| which proves the lemma.

Lemma 3.5. If g1(c)is the function of Lemma 3.4, then 2

XM i=1

|Si2|(|Si2| 1) =X

c2A

g1(c).

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Proof. Define an edge-colored graphGwith vertex setA, edge set[Mi=1S2i, and such that the color of edge {a, b}isa+b. The sumPM

i=1|Si2|(|Si2| 1) counts ordered pairs ({c, y},{x, z}) of distinct edges of G where {c, y} and {x, z} have the same color, i.e., c+y = x+z, and c, y, x, and z are all distinct elements of A. The sum P

c2Ag1(c) counts each such ordered pair ({c, y},{x, z}) exactly two times, one contribution coming from g1(c) and the other fromg1(y).

By Lemma 3.5,

XM i=1

|Si2|(|Si2| 1) 1 2

X

c2A

f(c). (9)

Next we use the following version of the Cauchy-Schwarz inequality.

Lemma 3.6. (Cauchy-Schwarz)Ifx1, . . . , xnare real numbers,t2{1,2, . . . , n 1}, and =1tPt

i=1xi 1 n

Pn

i=1xi, then Xn

i=1

x2i 1 n

Xn i=1

xi

!2

+tn 2 n t. A simple counting argument shows P

f(n) = |A|2 (|A| 2). Let P

c2Af(c) =

|A|2. If

:= 1

|A| X

c2A

f(c) 1 N

X

n

f(n) = |A| 1 N

X

n

f(n)

then, using Ruzsa’s bound|A|=O(N1/3) andC3+(N)F3+(N), we get

= |A|

|A|

2 (|A| 2)

N |A| C

whereC is some absolute constant. By Lemma 3.6, Xf(n)2

|A|2

2(|A| 2)2

N +|A| ·N( |A| C)2 N |A|

=

|A|

2

2(|A| 2)2

N + 2|A|3

⇣1 |A|C2

1 |A|N . By (7) and (9),

Xf(n)2  X

f(n) +|A| 2X

c2A

f(c) + XM i=1

|Si2|(|Si2| 1)

!

+ 96|A|2

 |A|3 2 +5|A|

2 X

c2A

f(c) + 96|A|2

= |A|3

✓1 + 5 2

+ 96|A|2.

(12)

Combining the two estimates onP

f(n)2 gives the inequality

|A|2

2(|A| 2)2

N + 2|A|3

⇣1 |A|C2

1 |A|N |A|3

✓1 + 5 2

+ 96|A|2. (10) If = 0, then (10) is not valid but we still get

|A|

2

2(|A| 2)2 N |A|3

2 + 96|A|2.

This inequality implies |A|(1 +o(1))(2N)1/3. Assume that >0. In this case, (10) simplifies to

|A|(1 +o(1)) 2 + 10 4 2 1/3N1/3. (11) At this point we find the maximum of the right hand side of (11) using the fact that 0 1 +|A|7 , which follows from Lemma 3.4. For|A| 28, the maximum occurs when = 1 +|A|7 therefore, after some simplifying, we find

|A|(1 +o(1))(8N)1/3.

4. Proof of Theorem 1.5(i)

The proof of Theorem 1.5(i) follows along the same lines as the proof of Theorem 1.3.

We will use the same notation as in the previous section. The derivation of (7) is very similar except in Z, or in ZN with N odd, there are fewer 3-term a.p.’s in A. Regardless, (7) still holds under the assumption that A ⇢[N] is aB3+-set, or A⇢ZN is aB3+-set andN odd.

Next we prove a lemma that corresponds to Lemma 3.2.

Lemma 4.1. If x2 Ti1\Tj1 for distinct i and j, then either si =sj = 2, or if sj >2, thensi= 2,sj = 3, and|Ti1\Tj1| 3.

Proof. The proof of this lemma is exactly the same as the proof of Lemma 3.2 up until the point where we write the equation 2z = 2u. In Z(or ZN with N odd), this impliesz=uwhich is a contradiction since the elementsx, bi1, y, z, u, vare all distinct. This allows us to conclude thatTi1\Tj1=;for anyi6=jwithsi 3 and sj 3.

The assertion|Ti1\Tj1| 3 can be verified with some easy computations. Alter- natively, one can just ignoreai3=uandbi3=vin Figure 1 to see|Ti1\Tj1| 3.

Corollary 4.2. If m+ 1i < jM, thenTi1\Tj1=;.

(13)

Proof. Ifx2Ti\Tj withi6=j, then by Lemma 4.1, one ofsi orsj must be equal to 2.

The next lemma has no corresponding lemma from the previous section. It will be used to estimateP

c2Af(c).

Lemma 4.3. IfA⇢[N]is a B3+-set or ifA⇢ZN is a B3+-set andN is odd, then for any a2A, the number of distinct i2{1,2, . . . , m}such that a2Ti1 is at most

|A|

2 .

Proof. To make the notation simpler, we suppose a 2 Ti1 for 1  i  k and we will show k  |A|2 . The case when a 2 Ti11 \· · ·\Ti1k for some sequence 1  i1 < · · · < ik  m is the same. For this lemma we deviate from the notation Si2 = {{ai1, bi1}, . . .{aisi, bisi}}. Write Si2 = {{a, ai},{bi, ci}} and a+ai =bi+ci where 1ik, and for fixedi, the elementsa, ai, bi, andciare all distinct. Observe a1, . . . , ak are all distinct since the sums a+ai are all distinct. For 1  i  k, a=bi+ci ai. Therefore,

bi+ci+aj =bj+cj+ai

for any 1i, j k. These two sums must intersect and they cannot intersect at aj or ai, unlessi=j, so for 2jk,

{b1, c1}\{bj, cj}6=;.

Let 2j l be the indices for which the sums intersect at b1. Letl+ 1j k be the indices for which the sums intersect atc1. Letb=b1 andc =c1. We have thekequations

a+a1 = b+c, a+a2 = b+c2,

... ... a+al = b+cl, a+al+1 = bl+1+c,

... ... a+ak = bk+c.

We will show thata1, . . . , ak, c1, . . . , cl, bl+1, . . . , bk are all distinct which implies 2k|A|.

Supposeai=bjfor some 2ilandl+1j k. Thena+bj=a+ai=b+ci, but a = bj +c aj. Therefore, b+ci = a+bj = 2bj+c aj which implies 2bj+c=b+ci+aj. The elementsaj, bj, andcare all distinct so these sums cannot

(14)

intersect ataj. Similarly they cannot intersect atc. The only remaining possibility isbj =ci, but thenai =bj=ci, which is a contradiction. We conclude thataiand bj are distinct for 2il and l+ 1j k. A similar argument shows thataj andci are distinct forl+ 1jkand 2il.

Suppose now that ai = ci0 for some 2  i 6= i0  l. Then b+ci = a+ai = a+ci0 =a+ (a+ai0 b), so that 2b+ci = 2a+ai0. Since 2i0 l, these sums cannot intersect atband they cannot intersect ata. Ifci=ai0, thena=bwhich is impossible. The equation 2b+ci= 2a+ai0 contradicts theB+3 property. Note that 2b= 2aneed not imply a=b ifA⇢ZN with N even. We conclude thatai 6=ci0

for each 2i6=i0 l. Similarly,aj 6=bj0 forl+ 1j6=j0k.

The previous two paragraphs imply

{a1, a2, . . . , ak}\{c2, c3, . . . , cl, bl+1, bl+2, . . . , bk}=;.

To finish the proof we show{c2, c3, . . . , cl}\{bl+1, bl+2, . . . , bk}=;. Supposeci=bj for some 2il andl+ 1jk. Then

a+ai=b+ci=b+bj =b+ (a+aj c) =b+a+aj (a+a1 b) =aj+ 2b a1

which implies a+ai +a1 = aj + 2b. Since i < l+ 1  j, these sums cannot intersect ataj. They cannot intersect atbeither sincea, ai, b, andciare all distinct whenever 1il. This is a contradiction. Therefore,ci6=bj for all 2iland l+ 1jk.

Lemma 4.4. If A⇢[N]is aB3+-set, then X

c2A

f(c)|A|2

2 + 4|A|.

Proof. Again we writef as a sum of the simpler functionsg1 andg2. Recall that forc2A,

g1(c) = #n

({x, z}, y)2A(2)⇥A:c=x y+z, c6=y,{x, z}\{y}=;o , and

g2(c) = #n

({x, z}, y)2A(2)⇥A:c=x y+z, c=y,{x, z}\{y}=;o . For eachc2A,f(c) =g1(c) +g2(c). The sumP

c2Ag2(c) is exactly the number of nontrivial 3-term a.p.’s inA. By Lemma 3.1, this is at most 4|A|.

Ifc /2T11[· · ·[TM1, then the equationc+y=x+zwithc, y, x, andzall distinct has no solutions inA sog1(c) = 0. Assumec2T11[· · ·[TM1.

Case 1: c /2T11[· · ·[Tm1.

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By Corollary 4.2, there is a uniquej withc2Tj1 andm+ 1jM. For such aj we have|Sj2||A|2 by Corollary 4.2. There is a unique pair inSj2that contains c so y is determined. There are at most |A|2 choices for the pair{x, z}2Sj2\{c, y} so g1(c)|A|2 .

Case 2: c2T11[· · ·[Tm1.

First assume c /2Tm+11 [· · ·[TM1. A solution toc+y=x+zwith c, y, x, and z all distinct corresponds to a choice of an Si2 with 1  i  m and c 2 Ti1. By Lemma 4.3,cis in at most |A|2 Ti1’s and sog1(c) |A|2 .

Lastly suppose c 2 Tm+11 [· · ·[TM1 . There is a unique j with c 2 Tj1 and m+ 1  j  M. Furthermore, for this j we have |Tj1| = 6 by Lemma 4.1. If c 2 Ti1 with 1 im then, again by Lemma 4.1, |Ti1\Tj1| 3. There are 63 3-subsets ofTj1 and given such a 3-subset, there are 31 ways to pair up an element in the 3-subset withcinS2i. This impliesc is in at most 3 63 Si2’s with 1im, so g1(c)  2 + 3 63|A|2 . The 2 comes from choosing one of the two pairs in Sj2\{c, y}.

The rest of the proof of Theorem 1.5(i) is almost identical to that of Theorem 1.3.

IfP

c2Af(c) = |A|2, then by (7) and (9), Xf(n)2|A|3

✓1 + 5 2

+O(|A|2).

We use the same version of the Cauchy-Schwarz inequality to get

|A|2

2(|A| 2)2

3N + 2|A|3

⇣1 C|A|⌘ 1 3N|A| |A|3

✓1 + 5 2

+O(|A|2). (12) If = 0, then

|A|

2

2(|A| 2)2 3N  |A|3

2 +O(|A|2)

which implies|A|(1 +o(1))(6N)1/3. Assume >0. Then (12) simplifies to

|A|(1 +o(1))(6 + 30 12 2)1/3N1/3.

By Lemma 4.4, 0  12+|A|3 . The maximum occurs when = 12+|A|3 and we get

|A|(1 +o(1))(18N)1/3.

If we were working inZN with N odd, then in (12) the 3N can be replaced by N. Some simple calculations show that we get Theorem 1.3 in the odd case. We actually obtain the upper bound|A|(1 +o(1))(6N)1/3whenA⇢ZNis aB3+-set andN is odd.

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5. Proof of Theorem 1.5(ii)

LetA⇢[N] be aB4+-set. Forn2[ 2N,2N], define

f(n) = #{({a1, a2},{b1, b2}) 2 A(2)⇥A(2) :a1+a2 b1 b2=n, {a1, a2}\{b1, b2}=;}.

Recall thatA(2)={{x, y}:x, y2A, x6=y}.

Lemma 5.1. If A⇢[N]is aB4+-set, then Ais a B2-set.

Proof. Suppose a+b = c+d with a, b, c, d 2 A. If {a, b}\{c, d}= ;, then the equation 2(a+b) = 2(c+d) contradicts the B+4 property so {a, b}\{c, d} 6= ;. Sincea+b=c+dand{a, b}\{c, d}6=;, we have{a, b}={c, d}.

Lemma 5.2. If A⇢[N]is aB4+-set, then for any integer n,f(n)2|A|.

Proof. Suppose f(n) 1. Fix a tuple ({a1, a2},{b1, b2}) counted by f(n). Let ({c1, c2},{d1, d2}) be another tuple counted byf(n), not necessarily di↵erent from ({a1, a2},{b1, b2}). Thena1+a2 b1 b2=c1+c2 d1 d2 so

a1+a2+d1+d2=c1+c2+b1+b2. (13) By the B4+ property, {a1, a2, d1, d2}\{c1, c2, b1, b2} 6= ;. In order for this in- tersection to be non-empty, it must be the case that {a1, a2}\{c1, c2} 6= ; or {b1, b2}\{d1, d2}6=;.

Case 1: {a1, a2}\{c1, c2}6=;.

Assumea1=c1. There are at most|A|choices forc2so we fix one. The equality a1=c1 and (13) imply

d1+d2=b1+b2+c2 a2. (14) The right hand side of (14) is determined. By Lemma 5.1, there is at most one pair {d1, d2}such that (14) holds.

Case 2: {a1, a2}\{c1, c2}=;and {b1, b2}\{d1, d2}6=;.

Again there is no loss in assumingb1=d1. There are at most|A| choices ford2 so fix one. The equality b1=d1 and (13) imply

c1+c2=a1+a2 b2+d2. (15) The right hand side of (15) is determined and there is at most one pair {c1, c2} satisfying (15) as before.

Putting the two possibilities together we get at most 2|A|solutions

({c1, c2},{d1, d2}). We have also accounted for the solution ({a1, a2},{b1, b2}) in our count sof(n)2|A|.

(17)

Lemma 5.3. If A⇢[N]is aB4+-set, then

Xf(n)(f(n) 1)2|A| X

n2A A

f(n). (16)

Proof. The left hand side of (16) counts the number of ordered tuples (({a1, a2},{b1, b2}),({c1, c2},{d1, d2}))

such that ({a1, a2},{b1, b2})6= ({c1, c2},{d1, d2}), and both tuples are counted by f(n). Equation (13) holds for these tuples. As before we consider two cases.

Case 1: {a1, a2}\{c1, c2}6=;.

Assumea1=c1 so thata2 c2=b1+b2 d1 d2.

If{b1, b2}\{d1, d2}6=;, sayb1=d1, thena2 c2=b2 d2. We can rewrite this equation asa2+d2=b2+c2so that{a2, d2}={b2, c2}. Since{a1, a2}\{b1, b2}=;, it must be the case that a2 =c2 and d2 =b2. This contradicts the fact that the tuples are distinct. We conclude{b1, b2}\{d1, d2}=;.

There are |A| choices for the element a1 =c1 and we fix one. Since a2 c2 = b1+b2 d1 d2 and{b1, b2}\{d1, d2}=;, there are f(a2 c2) ways to choose {b1, b2}and{d1, d2}. Also observe that each n2A A withn 6= 0 has a unique representation as n=a2 c2 witha2, c22A. This follows from the fact thatAis aB2-set.

Case 2: {a1, a2}\{c1, c2}=;and {b1, b2}\{d1, d2}6=;.

The argument in this case is essentially the same as that of Case 1.

Putting the two cases together proves the lemma.

Observe P

f(n) = |A|2 |A|2 2 . Using Cauchy-Schwarz, and Lemmas 5.3 and 5.2,

|A|

2 |A| 2 2

2

4N  X

f(n)2

✓|A| 2

◆✓|A| 2 2

+ 2|A| X

n2A A

f(n)

 |A|4

4 + 2|A||A A| ·2|A|

 |A|4

4 + 4|A|4=17|A|4 4 . After rearranging we get

|A|(1 +o(1))(16·17N)1/4= (1 +o(1))(272N)1/4.

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6. Proof of Theorem 1.6

Lemma 6.1. Let A be a Bk+-set with k 4. If k = 2l, then there is a subset A0 ⇢A such thatA0 is aB+l -set and|A0| |A| 2l. If k= 2l+ 1, then there is a subsetA0⇢Asuch that |A0| |A| 2kandA0 is either a Bl+-set or aBl+1+ -set.

Proof. Supposek = 2l with l 2. IfA is not a Bl+-set, then there is a set of 2l (not necessarily distinct) elementsa1, . . . , a2l2A, such that

a1+· · ·+al=al+1+· · ·+a2l

and {a1, . . . , al}\{al+1, . . . , a2l}=;. LetA0 =A\{a1, a2, . . . , a2l}. If A0 is not a B+l -set, then there is another set of 2lelements ofA0, sayb1, . . . , b2l, such that

b1+· · ·+bl=bl+1+· · ·+b2l

and{b1, . . . , bl}\{bl+1, . . . , b2l}=;. Adding these two equations together gives a1+· · ·+al+b1+· · ·+bl=al+1+· · ·+a2l+bl+1+· · ·+b2l

with {a1, . . . , al, b1, . . . , bl}\{al+1, . . . , a2l, bl+1, . . . , b2l}=;. This is a contradic- tion.

The case whenk= 2l+ 1 5 can be handled in a similar way.

It is easy to modify the proof of Lemma 6.1 to obtain a version forBk-sets.

Lemma 6.2. LetAbe aBk-set withk 4. Ifk= 2l, then there is a subsetA0⇢A such that A0 is a Bl-set and |A0| |A| 2l. If k = 2l+ 1, then there is a subset A0 ⇢Asuch that |A0| |A| 2k andA0 is either aBl-set or aBl+1-set.

ForA⇢[N] andj 2, let

j(n) = # (a1, . . . , aj)2Aj :a1+· · ·+aj=n . Lete(x) =e2⇡ix andf(t) =X

a2A

e(at). For anyj 1,f(t)j =P

j(n)e(nt) so by Parseval’s Identity,P

j(n)2=R1

0 |f(t)|2jdt. The next lemma is (5.9) of .

Lemma 6.3. If A⇢[N]is aBk-set, then X

k(n)2(1 +o(1))k2|A|X

k 1(n)2. (17)

In , Ruzsa estimates the right hand side of (17) using H¨older’s Inequality and shows

X

k 1(n)2⇣X

k(n)⌘kk 21

|A|k11. Our next lemma uses H¨older’s Inequality in a di↵erent way.

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