A NUMERICAL NOTE ON UPPER BOUNDS FOR B 2 [g] SETS

Transcription

1 A UMERICAL OTE O UPPER BOUDS FOR B [g] SETS Laurent Habsieger, Alain Plagne To cite this version: Laurent Habsieger, Alain Plagne. A UMERICAL OTE O UPPER BOUDS FOR B [g] SETS <hal v3> HAL Id: hal Submitted on 8 ov 016 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 A UMERICAL OTE O UPPER BOUDS FOR B [g] SETS LAURET HABSIEGER AD ALAI PLAGE Abstract. Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a satisfactory way. Their natural generalization, called B [g] sets and defined by the fact that there are at most g ways up to reordering the summands to represent a given integer as a sum of two elements of the set, are much more difficult to handle and not as well understood. In this article, using a numerical approach, we improve the best upper estimates on the size of a B [g] set in an interval of integers in the cases g =, 3, 4 and Introduction Let g be a positive integer. A set A of integers is said to be a B [g] set if for any integer n, there are at most g ways to represent n as a sum a+b with a, b A and a b. As is usual, we denote by F g, the largest possible size of a B [g] set contained in {0, 1,..., }. In the study of B [g] sets, this is the most studied aspect. These sets have a long history going back to Sidon [1] in the 30s and to the seminal work of Bose, Chowla and Singer as for lower bounds, and of Erdős and Turán as for upper bounds. See [13, 1,, 4]. Except in the case g = 1, for which it is known that F 1,, the precise asymptotic behaviour of F g, remains unknown. However, for any positive g, since a Sidon set is in particular a B [g] set, it is known that the quantity F g, grows at least like a constant times. Better lower bounds were obtained in [11, 7, 3, 9]. As for upper bounds, the current best result is 1 F g, min g, g 1, the first argument in the minimum being contained in [10] and the second one in [15]. otice that the first is better than the second as soon as g 6. otice finally that it is not even known whether there is a constant c g such that F g, c g. In this article, we slightly improve on the upper bound 1 for g < 6 by proving the following result. Theorem 1. One has F g, g 1. Both authors are supported by the AR grant Cæsar, number AR 1 - BS

3 LAURET HABSIEGER AD ALAI PLAGE For instance, we obtain F,.851 instead of Yu s.864. These improvements remain modest but one must remember that this is the case for all the recent ones since the beginning of the 000s when it was proved [5] that F,.913. Theorem 1 gives in particular a new best result in the cases g =, 3, 4 and 5 and corresponds to pushing Yu s method to some kind of extremity since it is not at all clear that the method is even able to prove F g, 1.74 g 1 although we conjecture that this is the case. We examine Yu s method and use an approach similar to the one used in [6] which led to new bounds for the dual problem of additive bases. In order to prove Theorem 1, we first reformulate Yu s method [14, 15] by giving a general explicit result Theorem depending on the choice of a fixed auxiliary function. This allows us to apply the method not only to the case of polynomials of high degree but directly to the case of power series. In this frame, Yu s bound corresponds to an appropriate choice of the auxiliary function. Finally, we optimize the use of our general result by computing numerically a best possible function of a certain type. This leads to Theorem 1.. The method Let A be a set of integers contained in {0, 1,..., }. We define the function ˆft = a A expπiat. In particular, ˆf0 = A. If d denotes the function counting the number of representations of an integer as a difference in A A, namely, for n Z, a,a A dn = {a, b A n : a b = n}, then we may compute that ˆft = expπia a t = dn expπint = dn cosπnt, where we use the parity of d which follows from the symmetry of A A as multiset. In this article, a function b will be called admissible if the following holds: its set of definition S b R is countable, symmetric with respect to zero and contains 0, b is an even function taking its values in the set of non negative real numbers R +, namely b : S b R + and, finally, θ S b bθ < +. In the sequel, for simplicity, we denote bθ = b θ. An admissible function b being chosen, we define the function w b as n w b t = θ S b b θ exp iπθt = θ S b b θ cos πθt,

4 SIDO 3 by parity of b. otice that the admissibility of b implies w b to be C R and even. Suppose that a subset A of {0, 1,..., } is given, as well as an admissible function b, we then define D A b = n dnw b. n It is easy to compute, interverting the order of summations, that D A b = dn b θ exp iπθ n n θ S b = iπθn b θ dn exp θ S b n = θ 3 b θ ˆf, θ S b the last equality following from. This shows in particular that D A b b 0 A Two lemmas In this section, we state two results which will be useful in our argument. The first one is a lemma of an analytical nature. If w is an even C R function, we denote I 1 w = I w = wt dt, wt dt, w = max t [0,1] w t, Aw = w 1 + w. Such an even C R function w being given, we define the function w as the unique - periodic function coinciding with w on [ 1, 1]. We have the following lemma. Lemma 1. Let w be an even C R function. For m Z, let Then we have a m = 1 1 wt = a Moreover, the following upper bound holds wt exp iπmt dt. m=1 a m Aw π m. a m cosπmt.

5 4 LAURET HABSIEGER AD ALAI PLAGE One also has I 1 w = a 0, I w = a I w I 1 w = Proof. Dirichlet s theorem ensures us that w coincides with its Fourier expansion. This is the first equality. As for the upper bound, we compute, using the parity of w, [ wt exp iπmt a m = iπm = 1 1 iπm = 1 π m = 1 π m 1 ] iπm w t exp iπmt dt + m=1 1 1 a m. [w t exp iπmt] m=1 a m, w t exp iπmt dt w t exp iπmt dt m w 1 w t exp iπmt dt 1 The upper bound of the lemma follows. Concerning the two first identities, they are immediately implied by standard calculus and the normal convergence of the Fourier series of w which allow to interchange summation and integration. The third one follows from the previous two. The second lemma is of an arithmetical nature. In the course of proving the Theorem, we shall meet the following quantity SA = 1 1 n ˆf A, n= where we use the notation of Section, in particular A is a set of integers included in {0, 1,..., }. The next lemma is an upper bound for SA. Lemma. If A is a B [g] set included in {0, 1,..., }, then SA g 1 A. Proof. It is for instance an intermediary result in the proof of Lemma 3 in [14]. The inequality follows from the fact that SA counts the number of solutions to the equation a b = c d with a, b, c, d A and a b..

6 SIDO 5 4. Proof of the Theorem We now come to the central estimate of this article which is an explicit version of Lemma. of [15]. With such a result, we can apply the method not only to the case of polynomials but also to the case of power series. Theorem. Let A be a B [g] set contained in {0, 1,..., } and b be an admissible function. We have D A b I 1 w b + Aw b A + w 4 b 0 I 1 w b A + I w b I 1 w b + Aw b 3/ g 1 A A 4 + A 3. Proof of Theorem. We apply Lemma 1 to w b which is C and even and use the notation introduced there for w b. By formula, we find D A b = n dnw b n = n dn w b n = + a 0 dn + πmn a m cos n = a 0 A + = a 0 = 1 ˆf0 + n= + m=1 a m m=1 n + + a n ˆf = w b 0 A + 1 = w b 0 A + 1 m=1 n + n= 1 n= a m ˆf πmn dn cos a n ˆf m a n + n A a n+k + a n k ˆf n A. Such a rearrangement of the terms of the series is allowed by the fact it is normally convergent. This follows from the bounds on the a m given by Lemma 1 and the boundedness of the terms ˆfn/ which are upper bounded by A.

7 6 LAURET HABSIEGER AD ALAI PLAGE Restarting from this identity on D A b, we obtain on recalling ˆf 0 = A 1 D A b = w b 0 A + 1 = w b 0 A n= a 0 + a n + a n + a n+k + a k + a n k ˆf A a k A a n+k + a n k ˆf n A n A and then D A b w b 0 A + 1 A a 0 + a k + a k A + 1 a n + a n+k + a n k ˆf n A. By the upper bound given in Lemma 1, we have a n+k, a n k Aw b π k 1 = Aw b 4 for n =,..., 1. We thus deduce D A b w b 0 A + a0 + Aw b A A a n + Aw b n ˆf A.

8 SIDO 7 The last term of this upper bound is bounded above using Cauchy-Schwarz inequality, more precisely a n + Aw b n ˆf A = a n ˆf n A + Aw b a n a n 1/ 1/ + Aw b 11/ + Aw b 3/ Plugging this bound, we finally obtain a0 D A b w b 0 A + + Aw b A A 4 1/ + 1 a n + Aw b 3/ w b 0 A a0 + Aw b 4 a n A A 1/ ˆf ˆf n ˆf n A ˆf n A n A 1/ A + Aw b 3/ SA A A 1/. It is now enough to use the final identities of Lemma 1. Since I 1 w b = a 0 / and 1 a n a n = I w b I 1 w b, we conclude using Lemma. n=1 Corollary 1. Let b be an arbitrary admissible function such that I 1 w b < 0, then we have F g, lim sup 1 g 1 I 1w b. I w b. 1/ 1/

9 8 LAURET HABSIEGER AD ALAI PLAGE Proof. Let A be a B [g] set in {1,..., } with A = F g,. In particular, A. For an arbitrary admissible function b, we apply Theorem and let tend to infinity. Since, by non-negativity of b and inequality 3, D A b 0, we obtain I 1 w b + o1 A I w b I 1 w b + o1 g 1 A A 4 + A 3. Thanks to the assumption that I 1 w b < 0, one can square the preceding inequality and we obtain I1 w b + o1 A 4 I w b I 1 w b g 1 A A 4 and, after simplification, The corollary follows. I w b + o1 A I w b I 1 w b g Optimization : Choosing b In view of Corollary 1, we are led to the optimization problem of computing max b admissible such that I 1 w b <0 I 1 w b I w b. In his paper [15], Yu first his Theorem 1 chooses the function M cos πm + λt wt = m + λ m=0 where M is taken equal to 10 6 and λ = 3/4 a case for which computations are made easier which gives the bound Yu then proceeds with a numerical optimization and finally, with λ = , he gets the value leading to Yu s second theorem this is the value mentioned in 1. There are several ways to improve on this result. First, our general result can be applied to any truncation of the infinite series which is non convergent for t = 0 associated with Yu s function. If we go back to the case λ = 3/4, and let M tend to infinity, this already gives the bound , which is the limit of Yu s function with this choice of parameter. But, again, one may then move slightly λ. We used a signed continued fraction method which leads us to consider the value λ = 365/478 at some step. With this choice of λ, we are led to the numerical upper bound We do not enter into more details here since this method does not give the best value we could obtain. In fact, there is no reason to choose such a regular function w. We started a numerical study on functions of the form wt = cos y 0 + πt + M j=1 c j j cos y j + j + 1πt.

10 SIDO 9 We used a Maple program and could go up to M = 400 that is, 801 variables. The computation took about four days on a shared machine equipped with two Intel Xeon E5-470v processors. otice that, more than time-consuming, this approach is very spaceconsuming and in fact limited by space considerations. In the above form, we were looking for an optimum where the y j are restricted to belong to 0, π and the c j to 0, 1. It turns out that when we increase the number M of variables, the values y j and c j seem to converge. Here are the first values that are given by the optimization process obtained for M = 400: and c 01 = , c 0 = , c 03 = , c 04 = , c 05 = , c 06 = , c 07 = , c 08 = , c 09 = , c 10 = , c 11 = , c 1 = , c 13 = , c 14 = , c 15 = , c 16 = , c 17 = , c 18 = , c 19 = , c 0 = , c 1 = , c = , c 3 = , c 4 = , c 5 = , c 6 = , c 7 = , c 8 = , c 9 = , c 30 = , c 31 = , c 3 = , c 33 = , c 34 = , c 35 = , c 36 = , c 37 = , c 38 = , c 39 = , c 40 = , c 41 = , c 4 = , c 43 = , c 44 = , c 45 = , c 46 = , c 47 = , c 48 = , c 49 = , c 50 = , y 00 = , y 01 = , y 0 = , y 03 = , y 04 = , y 05 = , y 06 = , y 07 = , y 08 = , y 09 = , y 10 = , y 11 = , y 1 = , y 13 = , y 14 = , y 15 = , y 16 = , y 17 = , y 18 = , y 19 = , y 0 = , y 1 = , y = , y 3 = , y 4 = , y 5 = , y 6 = ,

11 10 LAURET HABSIEGER AD ALAI PLAGE y 7 = , y 8 = , y 9 = , y 30 = , y 31 = , y 3 = , y 33 = , y 34 = , y 35 = , y 36 = , y 37 = , y 38 = , y 39 = , y 40 = , y 41 = , y 4 = , y 43 = , y 44 = , y 45 = , y 46 = , y 47 = , y 48 = , y 49 = , y 50 = The interested reader can refer to the complete numerical results available in [8]. otice that this function remains close to Yu s function, which after renormalization can be taken equal to wt = M m=0 λ cos m + 1π + λ 1πt. m + λ Indeed the coefficients c j remains around 0.75 while the coefficients y j are slightly above 1.5. Finally considering these values and those for bigger indices for M = 400 led us to the value and thus to Theorem 1. Heuristically, it seems that the method could be pushed up to proving the bound However, if true and provable by the present method, this could require to use a value of M much larger than 400. References [1] R. C. Bose, An affine analogue of Singer s theorem, J. Indian Math. Soc..S , [] S. Chowla, Solution of a problem of Erdős and Turán in additive number theory, Proc. at. Acad. Sci. India. Sect. A , 1. [3] J. Cilleruelo, I. Z. Ruzsa and C. Trujillo, Upper and lower bounds for finite B h [g] sequences, J. umber Theory 97 00, [4] P. Erdős and P. Turán, On a problem of Sidon in additive number theory and on some related problems, J. London Math. Soc , [5] B. Green, The number of squares and B h [g] sets, Acta Arith , [6] L. Habsieger, On finite additive -bases, Trans. Amer. Math. Soc , [7] L. Habsieger and A. Plagne, Ensembles B [] : l étau se resserre, Integers 00, A. [8] L. Habsieger and A. Plagne, [9] G. Martin and K. OBryant, Constructions of Generalized Sidon Sets, J. Combin. Theory Ser. A , [10] G. Martin and K. O Bryant, The supremum of autoconvolutions, with applications to additive number theory, Illinois Journal of Mathematics , o. 1, [11] A. Plagne, Recent progress on B h [g] sets, Congr. um , [1] S. Sidon, Ein Satz über trigonometrische Polynome und seine Anwendungen in der Theorie der Fourier- Reihen, Math. Annalen , [13] J. Singer, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc ,

12 SIDO 11 [14] Yu, An upper bound for B [g] sets, J. umber Theory 1 007, [15] Yu, A note on B [g] sets, Integers 8 008, A58. address: habsieger@math.univ-lyon1.fr Université de Lyon, CRS UMR 508, Université Claude Bernard Lyon 1, Institut Camille Jordan, 43 boulevard du 11 novembre 1918, 696 Villeurbanne Cedex, France address: plagne@math.polytechnique.fr Centre de Mathématiques Laurent Schwartz, Paris-Saclay, 9118 Palaiseau Cedex, France École polytechnique, CRS, Université