NEW UPPER BOUNDS FOR FINITE B h. Javier Cilleruelo

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1 NEW UPPER BOUNDS FOR FINITE B h SEQUENCES Javier Cilleruelo Abstract. Let F h N) be the maximum number of elements that can be selected from the set {,..., N} such that all the sums a + + a h, a a h are different. We introduce new combinatorial and analytic ideas to prove new upper bounds for F h N). In particular we prove F 3 N) π+) 4 N F 4 N) π+) 4 N /3 /4 + on /3 ), + on /4 ). Besides, our techniques have an independent interest for further research in additive number theory..introduction Let h be an integer. A subset A of integers is called a B h set if for every positive integer m, the equation m = x + + x h, x x h, x i A has, at most, one solution. Let F h N) denote the maximum number of elements that can be selected from the set {,..., N} so as to form a B h set. Bose and Chowla [] proved that F h N) N /h + on /h )..) 99 Mathematics Subject Classification. B3. Key words and phrases. B h sequences, Sidon sets. Typeset by AMS-TEX

2 JAVIER CILLERUELO On the other hand, if A is a B h subset of {,..., N}, an easy counting argument implies the trivial upper bound F h N) hh!n) /h..) The reader is refered to [9], [] and [5] for well written surveys about this topic. For h =, it is possible to take advantage of counting differences x i x j instead of sums x i + x j. In this way, P.Erdős and P.Turán [4] proved that F N) N / + ON /4 ), which is the best possible upper bound except for the estimate of the error term. Unfortunately, a similar argument doesn t work for h >. Denote by ma the set A+ +A = {a + +a m ; a i A}. In 969 Lindstrom [4] observed that if A is a B 4 sequence then A = A+A is nearly a B sequence. Using this idea he obtained the non-trivial upper bound for F 4 N). F 4 N) 8N) /4 + ON /8 )..3) X.D.Jia [0] generalized Lindstrom s argument to h = m and he obtained F m N) mm!) ) /m N /m + ON /4m )..4) An analytic proof of.4) has been given by M.Kolountzakis []. A similar upper bound for F m N) has been proved independently by S.Chen [] and S.W.Graham [7]: F m N) m!) ) /m ) N /m ) + ON /4m ) )..5) For large m m > 6 3 ), N.Alon unpublished) has obtained a better upper bound for F m N) exploiting the concentration of the sums a + + a m a a m around the value 0: F m N) 6 3/ mm!) ) /m N /m + on /m )..6) An sketch of the proof of.6) is contained in []. For m =, the estimate.5) gives F 3 N) 4N) /3 +ON /6 ). S.W.Graham [7], using an argument due to A.Li [3], made a slight improvement to this upper bound: F 3 N) 3.996N) /3 + O)..7)

3 NEW UPPER BOUNDS FOR FINITE B h SEQUENCES 3 In this paper we introduce new combinatorial and analytic ideas to improve all these upper bounds. Precisely, our aim is proving the following result. Theorem.. For 3 m < 38, For m 38, ) /3 4 F 3 N) + 6 N + on /3 ), π+) 4 ) /4 8 F 4 N) + 6 N + on /4 ). π+) 4 m!) ) /m ) F m N) + cos m π/m) N + on /m ) ), mm!) ) /m F m N) + cos m π/m) N + on /m ). F m N) ) /4 /m ) m!) N) + on /m ) ), 4m m F m N) ) ) /4 /m mm!) N + on /m ). 4m We shall use two different strategies to prove Theorem.. For small m m < 38) we will use that the sums a + + a m are not well distributed in the interval [m, mn]. It is the most interesting part of this paper. For large m m 38) we will take advantage of the concentration of the sums a + + a m a a m around the value 0. Section is devoted to prove a combinatorial identity for sequences of integers Lemma.) that has independent interest in additive number theory. See [4] and [5] for more applications of this identity.

4 4 JAVIER CILLERUELO In Section 3, inspired by an idea due to I.Ruzsa, we use Fourier analysis to prove that if A is a B m sequence contained in [, N], then the sequence ma [m, mn] is not well distributed in short intervals. A weaker result was used by I. Ruzsa, C.Trujillo and the author in [3] to obtain non trivial upper bounds for B h [g] sequences. In Section 4, we shall use the results of the previous sections to prove Theorem.. for the cases m < 38. Finally, in Section 5 we prove Theorem. for m 38, using a probabilistic approach.. A combinatorial lemma for sequences. In this section we present a combinatorial identity which works for general sequences of integers. Lemma.. Let A {,..., N}. Then, for any integer H we have h H d A h)h h) = H A N + H H A + N+H n= An) An H) where d A h) is number of solutions of h = a a ; counting function of A. ) H A, N + H a, a A and An) is the This identity captures valuable information about the sequence A: the size of A, the number of small differences a a and a measure of the distribution of the elements of A. Proof. An) An H) is the size of the set I n = {i; 0 i H, n A + i}. Then we can write the last sum of the lemma as where n N+H 0 i H I n H A N + H I n = n N+H n N+H I n + H A N + H, An) An H) = H A.

5 Then we have n N+H NEW UPPER BOUNDS FOR FINITE B h SEQUENCES 5 I n We also observe that ) H A = N + H n N+H I n H A N + H. I n = #{i, j); n A + i, n A + j} = #{i, j); n A + i) A + j)}. Then I n = A + i) A + j) = 0 n N+H 0 i,j H = H A + A + i) A + j). 0 i<j H It is easy to see that A + i) A + j) = A A + j i). Then A+i) A+j) = A A+h) H h) = d A h)h h), 0 i<j H h H h H and the lemma follows. The following corollary is an inmediate consequence of Lema.; however it loses the information about the distribution of the elements of A, which is crucial to improve.4) and.5). Corollary.. Let A [, N] a sequence of integers. Then for any integer H we have h H d A h)h h) H A N + H H A. S.W. Graham [7] deduced the above inequality from the Van der Corput lemma and applied it to the sequence ma when A is a B m sequence or a B m sequence to prove.4) and.5). A known fact about B m sequences is that most differences a + + a m a a m are different up to the order of the a i and a i. We state this fact see [7] for a proof) in a suitable form to apply it to Lemma. in Section 4.

6 6 JAVIER CILLERUELO Lemma.. Let A [, N] be a B m sequence. Then h H d ma h)h h) H + OHN m )/m ). Lemma.3. Let A [, N] be a B m sequence. Then h H d ma h)h h) H A m + OH A m ). In this paper we will take advantage of the last sum in Lemma.. It should be noted that this information is lost if we apply Corollary.. instead of lemma.. The sum N+H n= An) An H) H A N+H ) is a measure of the distribution of the elements of A in short intervals. We shall study this question in the next section. 3.The distribution of the elements of sumsets. The following theorem has an independent interest. In [3] we used it in a weaker version to obtain non trivial upper bounds for B h [g] sequences. Actually, the upper bounds obtained in [3] follow immediately from Theorem 3. by taking H = and µ = gm!. Given a sequence of integers A, we define r h n) = #{n = a + + a h ; a i A} and R h N) = n N r h n). Theorem 3.. Let A [, N] be a sequence of integers and h an integer. For any real µ and any positive integer H = on) we have where h n<hn+h L = R h n) R h n H) µ L h + o)) H A h, 4 π + ) and L h = cos h π/h) for h >. Proof. Let ft) = a A eiat. Then f h t) = h n hn r hn)e int, and for any m, 0 m H we have f h t)e imt = r h n m)e int. h n hn+h

7 Then f h t) = µ NEW UPPER BOUNDS FOR FINITE B h SEQUENCES 7 0 m H h n hn+h e imt = e int + h n hn+h h n hn+h for any real µ. Using the notation D k t) = k m=0 eimt we can write R h n) R h n H)) e int = R h n) R h n H) µ) e int f h t)d H t) =µe hit D hn )+H t) + R h n) R h n H) µ)e int. h n hn+h We known that sin k+ D k t) = t) sin t for 0 < t < π. 3.) Let t h = and we have π hn )+H. Then, for any integer j, 0 < j < N we have f h jt h )D H jt h ) = for any integer j, 0 < j < N. Taking absolute values gives fjt h ) h D H jt h ) D hn )+H jt h ) = 0. h n hn+h h n hn+h R h n) R h n H) µ) e injt h R h n) R h n H) µ = S. Now we use the fact that D k t) is a decreasing function for t, 0 t to conclude ) /h S fjt h ) 3.) D H Jt h ) for any integer j, j J N, whenever Jt h π H. π k+

8 8 JAVIER CILLERUELO In the next we will take J = [N/H) /3 ]. This choice implies that HJ H /3 N /3 H + N)/3, which gives Jt h < π/h. Now we are looking for a lower bound for fjt h ). Since the midpoint of the interval [, N] is N + )/, it will useful to express f as fjt h ) = e N+ ijt h f jt h ) where f jt h ) = a A N+ ia e )jt h. Suppose we have a function F x) = j J b j cosjx) such that F x) if x π h. Let C F = j b j. We are looking for a lower and an upper bound for J ) Re j= b jf jt h ). J J Re b j f jt h ) = Re b j e ia N+)/)jt h = j= a A = J b j cosa N +)/)jt h ) = F a N +)/)t h ) A, 3.3) a A j= a A because a N + )/)t h π h On the other hand, using 3.) we obtain J Re b j f jt h ) j= J b j f jt h ) = j= From this estimate and 3.3) we deduce S = h n hn+h j= for any integer a A. J ) /h S b j fjt h ) C F. D H Jt h ) j= R h n) R h n H) µ CF h D H Jt h ) A h. Now we obtain a lower bound for D H Jt h ) by noting that sin H D H Jt h ) = Jt h) sin Jt h H sin H Jt h) H Jt h H π 6 HJ N ) ),

9 NEW UPPER BOUNDS FOR FINITE B h SEQUENCES 9 where we have used the estimate sin x x H S + O N C h F x 6 ) /3 )) for x > 0. Then H A h. 3.4) To finish the theorem we look for a function F such that C F is as small as possible. For h > we choose F x) = cosπ/h) cos x with C F = cosπ/h). Then H ) )) /3 S cos h π/h) + O H A h N and we prove the theorem for h >. For the case h =, a little more work is required. If we consider the function it is easy to see that {, x π/ Gx) = + π cosx), π/ < x π Gx) = π cosx) + We need truncate the series. Let j= G J x) = π cosx) + J j= cosπj/) j cosπj/) j cosjx). cosjx). Obviously Gx) G J x) j> J 4j J ). We shall prove that the function F J x) = J J 3 G Jx) satisfies the suitable conditions. F J x) = J J 3 G Jx) = J J 3 Gx) + J J 3 G Jx) Gx)) and F J x) J J 3 Gx) J J 3 G Jx) Gx) J J 3 J J 3 J =

10 0 JAVIER CILLERUELO for x π/. On the other hand, C FJ = J π/ + J 3 J π/ + J 3 j= j J/ 4j 4j = J π/ + ). J 3 Substitution in 3.4) with J = [ N H )/3 ] gives the theorem for h =. S.W. Graham [8] has observed that the constant C F = π/ + is the best possible constant for h =. I include here his elegant proof. Proposition 3.. S.W. Graham) Suppose F x) = j b j cosjx) satisfies F x) for x π. Then C F = j b j π/ +. Proof. First note that F π/) = j b j ) j j b j. Next, we have 0 = π π π/ π F x)dx = F x)dx + F x)dx. 0 π/ Since F x) for x π/, it follows that π π/ F x)dx = j b j sinπj/) j π/, and this in turn gives us b j+ π/. j=0 So, we have lower bounds for the sum of b j, j even and j odd. Adding these bounds together the theorem folows. If we apply Theorem 3. to B m sequences we obtain the following corollary:

11 NEW UPPER BOUNDS FOR FINITE B h SEQUENCES Corollary 3.. If A [, N] is a B m sequence then, for any real µ and any integer H = on) where m n<mn+h D = ma)n) ma)n H) µ D m + o)) H A m, π + ) and D m = cosm π/m) m! for m >. Proof. Observe that if A is a B m sequence and s ma, then r m s) = m! except when the unique representation up to order) of s as a sum of m summands has some repeated summand. Then we can write r m s) = r ms) r ms) where r ms) = m! if s ma and r ms) = 0 if s ma. Obviously 0 r ms) r ms) m!. Let R mn) = k n r k) = m!ma)n) and R mn) = k n r k) Then R h n) R h n H) µm! = = m n mn+h m! m n mn+h m! ma)n) ma)n H) µ) R mn) R mn H)) m n mn+h + m n mn+h ma)n) ma)n H) µ R mn) R mn H). It should be noted that R mmn) counts the elements of ma with some repeated summand. Then R mn) R mn H) = HR mn) H A m. m n mn+h Now we apply Theorem 3. to get m! m n mn+h m n mn+h ma)n) ma)n H) µ R h n) R h n H) µm! + OH A m ) D m + o))h A m.

12 JAVIER CILLERUELO 4. Proof of Theorem. Cases m < 38 ) Let A be a B m or a B m sequence contained in [, N]. Lemma. to the sequence ma, which is contained in [, mn]. with µ = h H mn+h + n= d ma h)h h) = H ma mn + H H ma + ma)n) ma)n H) µ) 4..) Now we apply H ma mn+h. If we apply the Cauchy inequality to the last sum we obtain ma)n) ma)n H) µ) n mn+h mn + H ) n<mn+h ma)n) ma)n H) µ mn + H ) D m + o) ) H A m. We have applied Corollary 3. in the last inequality. Now, if h = m we use Lemma. to obtain an upper bound for the left hand side of 4.) giving H + OHN m )/m ) h H d ma h)h h) H ma mn + H H ma + mn + H ) D m + o))h A m. Trivially we have that A N /m and ma N /. Then, if we take H = [N /4m ] and divide the inequality by H we can write + o) ma mn + o)) o) + D m + o) mn + o)) A m. It should be noted that A is a priori a B m sequence. Then ) A + m ma = A m m m!

13 and, consequently, so, finally, and NEW UPPER BOUNDS FOR FINITE B h SEQUENCES 3 mn + o)) A m m!) + D m + o)) A m A m!) /m m + o)) N /m. + Dm The numbers D m are defined in Corollary 3. and we obtain the upper bounds ) /4 F 4 N) N + on /4 ) 7.8N) /4 + on /4 ) π+) 4 F m N) m /m m!) /m + cos m π/m)) /m N /m + on /m ), m >. If h = m we use Lemma.3 to get an upper bound for the left side of 4.) and we proceed in a similar way. H A m + OH A m ) h H d ma h)h h) H ma mn + H H ma + mn + H ) D m + o))h A m. Trivially we have that A N /m ) and ma N m/m ). If we take H = [N / A m ] and divide the inequality by H A we can write m + o) ma A mn + o)) o) + D m + o) N + o)) A m. Again, it should be noted that A is a priori a B m sequence. Then ) A + m ma = A m m m! and N + o)) A m m!) + D m + o)) A m,

14 4 JAVIER CILLERUELO so, finally, A m!) /m ) + o)) N /m ). + Dm Again, the numbers D m are defined in Corollary 3. and we obtain the upper bounds ) /3 F 3 N) N + on /3 ) 3.9N) /3 + on /3 ), π+) 4 F m N) m!) /m + cos m π/m)) /m N /m +on /m ) ), m >. 5. Proof of Theorem. Cases m 38). A probabilistic approach. Suppose that A [, N] is a B m sequence. Let the random variable Y be defined by Y = X + + X m X X m, where the X j are independent random variables uniformly distributed in A. We will take advantage of the concentration of Y around the value 0, considering the 4 moment of Y. We know that E 4 = EY 4 ) λ 4 N 4 P Y λn). Then P Y < λn) E 4 λ 4 N 4 5.) On the other hand we have that P Y < λn) A m rn), n <λn where rn) = #{n = a + + a m a a m; a i, a i A}. Let rn) = r n) + r n), where r n) counts the representations where all the a i, a i are distinct and r n) counts the representations with some repeated element.

15 NEW UPPER BOUNDS FOR FINITE B h SEQUENCES 5 Then, r n) A m N m m < on) because of the trivial estimate A N /m. Now we observe that if A is a B m sequence, the representation n = a + + a m a a m, a < < a m, a < < a m is unique. So r n) m!) due to the m! permutations of the a i and a i. and Now we can write n <λn rn) λnm!) + on). 5.) P Y < λn) A m λ m!) + o) ) N. 5.3) From 5.) and 5.3) we deduce A m and taking λ = 5E 4 N 4 ) /4 we have A m 5 λ 5 N 4 λ 4 N 4 E 4 m!) + o) ) N ) /4 5E4 m!) N 4 + o) ) N 5.4) The next step is to obtain an upper bound for E 4 = E Y 4 ). It is useful to write Y = Y + + Y m, where Y i = X i X i. The first observation is that EY j i ) = 0 if j is odd. Then EY + + Y m ) 4 ) = 4! γ + +γ m =4 = mey 4 ) + 3mm )E Y ) Observe that EY 4 ) N EY ) and EY γ ) EY γ m m ) γ! γ m! EY ) = EX X ) ) = EX ) E X ) EX )N EX ))) N. Then E 4 N 4 m ) 3mm ) + 4 = 3m m N 4 4

16 6 JAVIER CILLERUELO If we substitute in 5.4) we obtain A m 5 ) /4 5 3 /m) mm!) + o) ) N. 4 and we finish the proof of Theorem.. for B m sequences. If A is a B m sequence the proof is similar except that in this case we have the following estimate in 5.): n <λn rn) A m λm!) + o) ) N and the proof follows in the same way. Acknowledgements. I am indebted to I.Ruzsa for many valuable suggestions and comments. Thanks are also due to S.W.Graham who allowed me to include here Proposition 3.. and to M.Kolountzakis who mentioned to me the unpublished result of N.Alon. References. R.C.Bose and S.Chowla, Theorems in the additive theory of numbers, Comment. Math. Helv /963), S.Chen, On the size of finite Sidon sequences, Proc. Amer. Math. Soc. 994), J.Cilleruelo, I.Ruzsa and C.Trujillo, Upper and lower bounds for B h [g] sequences, to appear in J. Number Theory. 4. J. Cilleruelo and G.Tenenbaum, An overlapping lemma and applications, preprint. 5. J.Cilleruelo, Gaps in dense Sidon sets, preprint. 6. P.Erdős and P.Turan, On a problem of Sidon in additive number theory and on some related problems, J.London Math.Soc. 6 94), -5;P. Erdős, Addendum 9 944), S.W.Graham, B h sequences, Proceedings of a Conference in Honor of Heini Halberstam B.C.Berndt, H.G.Diamond, A.J.Hildebrand, eds.), Birkhauser, 996, pp S.W.Graham, Personal communication. 9. H.Halberstam and K.F.Roth, Sequences, Springer-Verlag, New York 983). 0. X.D.Jia, On finite Sidon sequences, J. Number Theory ), M.Kolountzakis, Problems in the Additive Number Theory of General Sets, I. Sets with distinct sums., Avalaible at kolount/surveys.htlm 996).. M.Kolountzakis, The density of B h [g] sequences and the minimum of dense cosine sums, J. Number Theory ), An Ping Li, On B 3 sequences Chinese), Acta Math. Sinica 34 99), B.Lindstrom, A remark on B 4 sequences, J. Comb. Theory 7 969),

17 NEW UPPER BOUNDS FOR FINITE B h SEQUENCES 7 5. A.Sárkőzy and V.T.Sós, On additive representation functions, The mathematics of Paul Erdős, Algorithms and Combinatorics 3 R.L. Graham and J. Nesetril, eds.), Springer- Verlag, New York Heidelberg Berlin, pp Departamento de Matematicas. Universidad Autónoma de Madrid. Madrid SPAIN address: franciscojavier.cilleruelo@uam.es