Javier Cilleruelo. Departamento de Matemáticas. Universidad Autónoma de Madrid MADRID ESPAÑA

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1 B [g]sequences WHOSE TERMS ARE SQUARES Javier Cilleruelo Departamento de Matemátias Universidad Autónoma de Madrid MADRID ESPAÑA INTRODUCTION Sixty years ago Sidon [7] asked, in the ourse of some investigations of Fourier series, for a sequene a < a... for whih the sums a i + a j are all distint and for whih a k tends to infinity as slowly as possible. Sidon alled these sequenes, B sequenes. The greedly algorithm gives a k k 3 and this was the best result until Atjai, Komlos and Szemeredi [] found a B sequene satisfying a k = ok 3 ). However, this result is far from the main onjeture about B sequenes. Conjeture. Corresponding to every ɛ > 0, there exists a B sequene A suh that a j j +ɛ. In general we say that a sequene A is a B [g] sequene if r n A) g for all integer n, where r n A) is the number of representations of n in the form n = a + b, a b a, b A). In 960, P.Erdös and A.Renyi [5], using probabilisti methods, proved the following first steps towards the onjeture. Theorem. Erdös-Renyi): Corresponding to every ɛ > 0, there exists a natural number g and a B [g] sequene A suh that a j j +ɛ. On the other hand it is easy to see that a B [g] sequene has to satisfy a j j. Thus, it seems that the sequene of squares is on the border line for this kind of problem. For this reason and beause of the onexion between the additive properties of the sequene of squares and the Fourier Series in the form a k e ikx, we have been interested in the study of B sequenes whose terms are squares. The entire sequene of squares annot be a B [g] sequene for any g beause the funtion rn) = #{n; n = a + b, a b} is not bounded uniformily in n. However, in [3] we proved the existene of a B -sequene of squares {a k } suh that a k k4. The reader is referred also to [4] and to the disussion in [6].) The main purpose of this note is to show that A in the Erdös-Renyi theorem an be taken to be a subsequene of the squares.

2 Theorem. Corresponding to every ɛ > 0, there exists a natural number g and a B [g] sequene of squares {a k } suh that a k k +ɛ. We will use Erdös onstrution of a probability measure on the spae of integer sequenes suh that in the resulting probability spae) almost all integer sequenes have some presribed rate of growth; thus the probable behaviour of the representation funtion in the addtion of sequenes of preribed rates of growth may then be investigated without further referene to these rates of growth. PROOF OF THE THEOREM. We will prove the theorem showing, for every ɛ > 0, the existene of a natural number g and a sequene {a k }, with a k k + ɛ suh that for every integer n nɛ), the number of the representations of n in the form n = a k + a j, a k a j is less or equal than g. Let Ω = {ω} be the spae of all the sequenes of integers. First of all we will onstrut, for every ɛ > 0, a probability spae suh that with probability the sequenes in that spae satisfy a j ɛ j + ɛ. We will need the following two theorems whih an be found in [6], pg Theorem. Let α, α, α 3,... be real numbers satisfying 0 α n. Then. there exists a probability spae Ω, S, µ) with the following two properties: i) For every natural number n, the event B n) = {ω; n ω} is measurable and µb n) ) = α n. ii) The events B ), B ),... are independents. Theorem. Let α j = j for every intege, 0 < <. Then, with probability in the spae desribed above, the elements a j of the sequenes ω = {a j } satisfy a j )j as j. For our purpose we hoose = ɛ in the latter theorem. Then, with probability, the elements a j of the sequenes ω = {a j } satisfy a j +ɛ j+ ɛ. + ɛ We define r n ω) = #{n = a j + a k ; a j a k, a j, a k ω}. Next, we will prove that for every g > = ɛ, with probability, r nω) g for every integer n nɛ). Erdös and Renyi also obtained the estimation g > ɛ. We appeal to the Borel-Cantelli lemma. Theorem. Let {E n } be a sequene of measurable events. If µe n ) < + ; then, with probability, at most a finite number of suh n= events an our.

3 For a natural number g > we onsider the events E n = {ω; r n ω) > g} and we will prove that µe n ) < +. Then, with probability, at most a finity n= number of events an our and the theorem will follow. We have µe n ) = d g+ µe n,d ), where E n,d = {ω; r n ω) = d}. Let rn) be the funtion rn) = #{n = a + b ; 0 < a b}. Then, n = a + b = = a rn) + b rn), a i b i. If r n ω) = d then eah omponent of exatly d pairs a j, b j ),..., a jd, b jd ) among the rn) pairs a, b ),..., a rn), b rn) ) belongs to ω. Let E n,d j,..., j d ) be the event 3 {ω; a j, b j,..., a jd, b jd ω, a k or b k ω, k j i, i =,..., d}. Then µe n,d ) = µe n,d j,..., j d )) j <j d rn) and µe n,d j,..., j d )) = d µ{ω; a ji, b ji ω} i= k j i i=,...,d µ{ω; a k, b k ω}); here µ{ω; a, b ω} = a b exept when a = b. In this ase µ{ω; a, a ω} = µ{ω; a ω} = a. Estimation of µe n ) for n a. We have whene µe n,d ) = j < <j d rn) µe n,d j,..., j d )) d i= µe n,d j,..., j d )) a ji b ji ), d j < <j d rn) i= a ji b ji )

4 4 d! a +b =n ab ab) d. If a + b = n, a b, then b n/. We define a n = µe n,d ) d! rn) a n n/) It is a well known fat that, for every δ > 0, rn) exists n 0 = n 0 ɛ) suh that for n > n 0. µe n ) = d g+ n δ ) d. min a and we have a +b =n,a>0 0 as n. Hene there rn) a n n/) for every n > n 0, and we have µe n,d ) Estimation of µe n ) for n = a. Here for > n 0. rn) g + )! a n n/) µ{ω; r n ω) > g} = µ{ω; r n ω) > g, a ω} + µ{ω; r n ω) > g, a ω} rn) a g! a n n/) ) g + a ) 4 rn) g + )! n/) rn) g + )! a n n/) Completion of proof. Define = max rn), and suppose 0 < δ < j n< j+ 4. From above, < δj for j 0 > jδ), and we may suppose also that j 0 n 0.

5 5 We write µe n ) = µe n ) + µe n ). n= n j 0 n> j 0 The first sum is finite and the seond sum an be written in the form say; at one j>j 0 m=g+ Σ j>j 0 j j+ rn)=m,n=a m=g+ µe n ) + 4 rj g + )! j 4 g + )! and this sum is finite if g > + 4δ. δ Next, Σ r j j>j 0 m=g+ g + )! j>j 0 m=g+ k j j j+ rn)=m,n a µe n ) = Σ + Σ, #{n = a ; j n < j+ } j n< j+ rn)=m,n a k j ) jδg+) j>j 0 jg+) j. g + )! a n j ) #{n; j n < j+, a n = k} g + )! jδg+) j>j 0 jg+) j k k g+). The last serie is onvergent beause g >. Then Σ is finite if g satisfies g > + 4δ. δ Finally, it is lear that δ an be hosen small enough so that the natural number g > satisfies even the last ondition.

6 6 Aknownledgment. Thank Professor Erdös for his advise. REFERENCES. [] Atjai, Komlos and Szemeredi, On finite Sidon sequenes, European J.Comb. 980), -. [] A.O.L.Atkin, On pseudo squares. Pro. Lond. Math. So. 4 A 965). [3] J.Cilleruelo, B sequenes whose terms are squares. Ata Arithmetia. LV 990). [4] J.Cilleruelo and A.Córdoba, B [ ] sequenes of squares. Ata Arithmetia. LXI.3 99). [5] P.Erdös and A.Renyi, Additive properties of radom sequenes of positive integers. Ata Arithmetia ) [6] H.Halberstam and K.F.Roth, Sequenes. Oxford Univ. Press. 966). [7] B.Sidon, Ein Satz über trigonometrishe Polynome und seine Anwendung in der Theorie der Fourier-Reihen. Math. Annln ).

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