International audienceSidon 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 2 [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 2 [g] set in an interval of integers in the cases g = 2, 3, 4 and 5

Habsieger, Laurent

Plagne, Alain

HAL-Polytechnique

A NUMERICAL NOTE ON UPPER BOUNDS FOR B2 [g] SETSLaurent Habsieger, Alain PlagneTo cite this version:Laurent Habsieger, Alain Plagne. A NUMERICAL NOTE ON UPPER BOUNDS FOR B 2[g] SETS. 2016. <hal-01361725v3>HAL Id: hal-01361725https://hal.archives-ouvertes.fr/hal-01361725v3Submitted on 8 Nov 2016HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.A NUMERICAL NOTE ON UPPER BOUNDS FOR B2[g] SETSLAURENT HABSIEGER AND ALAIN PLAGNEAbstract. Sidon sets are those sets such that the sums of two of its elements nevercoincide. They go back to the 30s when Sidon asked for the maximal size of a subset ofconsecutive integers with that property. This question is now answered in a satisfactoryway. Their natural generalization, called B2[g] sets and defined by the fact that there areat most g ways (up to reordering the summands) to represent a given integer as a sum oftwo 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 thesize of a B2[g] set in an interval of integers in the cases g = 2, 3, 4 and 5.1. IntroductionLet g be a positive integer. A set A of integers is said to be a B2[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,N) the largest possible size of a B2[g] set contained in {0, 1, . . . , N}. Inthe study of B2[g] sets, this is the most studied aspect.These sets have a long history going back to Sidon [12] in the 30s and to the seminalwork of Bose, Chowla and Singer as for lower bounds, and of Erdo˝s and Tura´n as for upperbounds. See [13, 1, 2, 4]. Except in the case g = 1, for which it is known thatF (1, N) ∼√N,the precise asymptotic behaviour of F (g,N) remains unknown.However, for any positive g, since a Sidon set is in particular a B2[g] set, it is knownthat the quantity F (g,N) grows at least like a constant times√N . Better lower boundswere obtained in [11, 7, 3, 9]. As for upper bounds, the current best result is(1) F (g,N) .√min(3.1694 g, 1.74217 (2g − 1))N,the first argument in the minimum being contained in [10] and the second one in [15].Notice that the first is better than the second as soon as g ≥ 6. Notice finally that it isnot even known whether there is a constant cg such that F (g,N) ∼ cg√N .In this article, we slightly improve on the upper bound (1) for g < 6 by proving thefollowing result.Theorem 1. One hasF (g,N) .√1.740463 (2g − 1) N.Both authors are supported by the ANR grant Cæsar, number ANR 12 - BS01 - 0011.12 LAURENT HABSIEGER AND ALAIN PLAGNEFor instance, we obtain F (2, N) . 2.2851√N instead of Yu’s 2.2864√N . These im-provements remain modest but one must remember that this is the case for all the recentones since the beginning of the 2000s when it was proved [5] that F (2, N) . 2.2913√N .Theorem 1 gives in particular a new best result in the cases g = 2, 3, 4 and 5 and corre-sponds to pushing Yu’s method to some kind of extremity since it is not at all clear thatthe method is even able to prove F (g,N) .√1.74 (2g − 1) N (although we conjecturethat this is the case).We examine Yu’s method and use an approach similar to the one used in [6] which led tonew bounds for the dual problem of additive bases. In order to prove Theorem 1, we firstreformulate Yu’s method [14, 15] by giving a general explicit result (Theorem 2) dependingon the choice of a fixed auxiliary function. This allows us to apply the method not onlyto the case of polynomials (of high degree) but directly to the case of power series. In thisframe, 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 functionof a certain type. This leads to Theorem 1.2. The methodLet A be a set of integers contained in {0, 1, . . . , N}. We define the functionfˆ(t) =∑a∈Aexp(2piiat).In particular, fˆ(0) = |A|.If d denotes the function counting the number of representations of an integer as adifference in A−A, namely, for n ∈ Z,d(n) = |{(a, b) ∈ A2 : a− b = n}|,then we may compute that(2) |fˆ(t)|2 =∑a,a′∈Aexp(2pii(a− a′)t) =∑|n|≤Nd(n) exp(2piint) =∑|n|≤Nd(n) cos(2pint),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 ofdefinition Sb ⊂ R is countable, symmetric with respect to zero and contains 0, b is an evenfunction taking its values in the set of non negative real numbers R+, namely b : Sb → R+and, finally, ∑θ∈Sbb(θ) < +∞.In the sequel, for simplicity, we denote b(θ) = bθ.An admissible function b being chosen, we define the function wb aswb(t) =∑θ∈Sbbθ exp (2ipiθt) =∑θ∈Sbbθ cos (2piθt) ,SIDON 3by parity of b. Notice that the admissibility of b implies wb to be C∞(R) and even.Suppose that a subset A of {0, 1, . . . , N} is given, as well as an admissible function b,we then defineDA(b) =∑|n|≤Nd(n)wb( nN).It is easy to compute, interverting the order of summations, thatDA(b) =∑|n|≤Nd(n)∑θ∈Sbbθ exp(2ipiθnN)=∑θ∈Sbbθ∑|n|≤Nd(n) exp(2ipiθnN)=∑θ∈Sbbθ∣∣∣∣fˆ ( θN)∣∣∣∣2 ,(3)the last equality following from (2). This shows in particular that DA(b) ≥ b0|A|2 ≥ 0.3. Two lemmasIn this section, we state two results which will be useful in our argument. The first oneis a lemma of an analytical nature. If w is an even C2(R) function, we denoteI1(w) =∫ 10w(t) dt,I2(w) =∫ 10w(t)2 dt,‖w′′‖ = maxt∈[0,1]|w′′(t)|,A(w) = |w′(1)|+ ‖w′′‖.Such an even C2(R) function w being given, we define the function w˜ as the unique 2-periodic function coinciding with w on [−1, 1].We have the following lemma.Lemma 1. Let w be an even C2(R) function. For m ∈ Z, letam =∫ 1−1w(t) exp(−ipimt) dt.Then we havew˜(t) =a02++∞∑m=1am cos(pimt).Moreover, the following upper bound holds|am| ≤ 2A(w)pi2m2.4 LAURENT HABSIEGER AND ALAIN PLAGNEOne also hasI1(w) =a02,I2(w) =a204+12+∞∑m=1a2m,2(I2(w)− I1(w)2) =+∞∑m=1a2m .Proof. Dirichlet’s theorem ensures us that w˜ coincides with its Fourier expansion. This isthe first equality.As for the upper bound, we compute, using the parity of w,am =[w(t) exp(−ipimt)−ipim]1−1+1ipim∫ 1−1w′(t) exp(−ipimt) dt=1ipim∫ 1−1w′(t) exp(−ipimt) dt=1pi2m2([w′(t) exp(−ipimt)]1−1 −∫ 1−1w′′(t) exp(−ipimt) dt)=1pi2m2(2(−1)mw′(1)−∫ 1−1w′′(t) exp(−ipimt) dt).The upper bound of the lemma follows.Concerning the two first identities, they are immediately implied by standard calculusand the normal convergence of the Fourier series of w˜ which allow to interchange summationand 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 quantityS(A) = 12NN−1∑n=−N(∣∣∣fˆ ( n2N)∣∣∣2 − |A|)2 ,where we use the notation of Section 2, in particular A is a set of integers included in{0, 1, . . . , N}. The next lemma is an upper bound for S(A).Lemma 2. If A is a B2[g] set included in {0, 1, . . . , N}, thenS(A) ≤ (2g − 1)|A|2.Proof. It is for instance an intermediary result in the proof of Lemma 3 in [14]. Theinequality follows from the fact that S(A) counts the number of solutions to the equationa− b = c− d with a, b, c, d ∈ A and a 6= b. SIDON 54. Proof of the TheoremWe now come to the central estimate of this article which is an explicit version of Lemma2.2 of [15]. With such a result, we can apply the method not only to the case of polynomialsbut also to the case of power series.Theorem 2. Let A be a B2[g] set contained in {0, 1, . . . , N} and b be an admissible func-tion. We haveDA(b) ≤(I1(wb) +A(wb)4N2)|A|2 + (wb(0)− I1(wb)) |A|+(√2 (I2(wb)− I1(wb)2) + A(wb)2N3/2)√(2g − 1)N |A|2 − |A|42+ |A|3 .Proof of Theorem 2. We apply Lemma 1 to wb (which is C2 and even) and use the notationintroduced there for w˜b. By formula (2), we findDA(b) =∑|n|≤Nd(n)wb( nN)=∑|n|≤Nd(n)w˜b( nN)=∑|n|≤Nd(n)(a02++∞∑m=1am cos(pimnN))=a02|A|2 ++∞∑m=1am∑|n|≤Nd(n) cos(pimnN)=a02∣∣∣fˆ(0)∣∣∣2 + +∞∑m=1am∣∣∣fˆ ( m2N)∣∣∣2=12+∞∑n=−∞an∣∣∣fˆ ( n2N)∣∣∣2= wb(0)|A|+ 12+∞∑n=−∞an(∣∣∣fˆ ( n2N)∣∣∣2 − |A|)= wb(0)|A|+ 12N−1∑n=−N(an +∞∑k=1an+2kN +∞∑k=1an−2kN)(∣∣∣fˆ ( n2N)∣∣∣2 − |A|) .Such a rearrangement of the terms of the series is allowed by the fact it is normallyconvergent. This follows from the bounds on the am given by Lemma 1 and the boundednessof the terms |fˆ(n/2N)| (which are upper bounded by |A|).6 LAURENT HABSIEGER AND ALAIN PLAGNERestarting from this identity on DA(b), we obtain (on recalling fˆ (0) = |A| ≥ 1)DA(b) = wb(0)|A|+ 12N−1∑n=−N(an +∞∑k=1an+2kN +∞∑k=1an−2kN)(∣∣∣fˆ ( n2N)∣∣∣2 − |A|)= wb(0)|A|+ 12(a0 +∞∑k=1a2kN +∞∑k=1a−2kN)(|A|2 − |A|)+12∑n=−N,...,N−1n 6=0(an +∞∑k=1an+2kN +∞∑k=1an−2kN)(∣∣∣fˆ ( n2N)∣∣∣2 − |A|)and thenDA(b) ≤ wb(0)|A|+ 12(a0 +∣∣∣∣∣∞∑k=1a2kN∣∣∣∣∣+∣∣∣∣∣∞∑k=1a−2kN∣∣∣∣∣) (|A|2 − |A|)+12∑n=−N,...,N−1n 6=0(|an|+∣∣∣∣∣∞∑k=1an+2kN∣∣∣∣∣+∣∣∣∣∣∞∑k=1an−2kN∣∣∣∣∣) ∣∣∣∣∣∣∣fˆ ( n2N )∣∣∣2 − |A|∣∣∣∣ .By the upper bound given in Lemma 1, we have∣∣∣∣∣∞∑k=1an+2kN∣∣∣∣∣ ,∣∣∣∣∣∞∑k=1an−2kN∣∣∣∣∣ ≤∞∑k=12A(wb)pi2((2k − 1)N)2 =A(wb)4N2for n = −N, . . . , N − 1. We thus deduceDA(b) ≤ wb(0)|A|+(a02+A(wb)4N2)(|A|2 − |A|)+12∑n=−N,...,N−1n6=0(|an|+ A(wb)2N2) ∣∣∣∣∣∣∣fˆ ( n2N )∣∣∣2 − |A|∣∣∣∣ .SIDON 7The last term of this upper bound is bounded above using Cauchy-Schwarz inequality,more precisely∑n=−N,...,N−1n 6=0(|an|+ A(wb)2N2) ∣∣∣∣∣∣∣fˆ ( n2N )∣∣∣2 − |A|∣∣∣∣=∑n=−N,...,N−1n6=0|an|∣∣∣∣∣∣∣fˆ ( n2N )∣∣∣2 − |A|∣∣∣∣+ A(wb)2N2 ∑n=−N,...,N−1n 6=0∣∣∣∣∣∣∣fˆ ( n2N )∣∣∣2 − |A|∣∣∣∣≤ ∑n=−N,...,N−1n 6=0|an|21/2+A(wb)2N2(2N − 1)1/2 ∑n=−N,...,N−1n6=0∣∣∣∣∣∣∣fˆ ( n2N )∣∣∣2 − |A|∣∣∣∣21/2≤ ∑n=−N,...,N−1n 6=0|an|21/2+A(wb)√2N3/2 ∑n=−N,...,N−1n 6=0∣∣∣∣∣∣∣fˆ ( n2N )∣∣∣2 − |A|∣∣∣∣21/2.Plugging this bound, we finally obtainDA(b) ≤ wb(0)|A|+(a02+A(wb)4N2)(|A|2 − |A|)+12 ∑n=−N,...,N−1n6=0|an|21/2+A(wb)√2N3/2 ∑n=−N,...,N−1n6=0∣∣∣∣∣∣∣fˆ ( n2N )∣∣∣2 − |A|∣∣∣∣21/2≤ wb(0)|A|+(a02+A(wb)4N2)(|A|2 − |A|)+12∑n=−N,...,N−1n 6=0|an|21/2+A(wb)2N3/2(NS(A)− (|A|2 − |A|)22)1/2.It is now enough to use the final identities of Lemma 1. Since I1(wb) = a0/2 and12∑n=−N,...,N−1n 6=0|an|2 ≤∞∑n=1|an|2 = 2(I2(wb)− I1(wb)2),we conclude using Lemma 2. Corollary 1. Let b be an arbitrary admissible function such that I1(wb) < 0, then we havelim supN→∞F (g,N)2(2g − 1)N ≤ 2(1− I1(wb)2I2(wb)).8 LAURENT HABSIEGER AND ALAIN PLAGNEProof. Let A be a B2[g] set in {1, . . . , N} with |A| = F (g,N). In particular, |A| √N .For an arbitrary admissible function b, we apply Theorem 2 and let N tend to infinity.Since, by non-negativity of b and inequality (3), DA(b) ≥ 0, we obtain(−I1(wb) + o(1)) |A|2 ≤(√2 (I2(wb)− I1(wb)2) + o(1))√(2g − 1)N |A|2 − |A|42+ |A|3 .Thanks to the assumption that I1(wb) < 0, one can square the preceding inequality andwe obtain (I1(wb)2 + o(1)) |A|4 ≤ 2 (I2(wb)− I1(wb)2)((2g − 1)N |A|2 − |A|42)and, after simplification,(I2(wb) + o(1)) |A|2 ≤ 2(I2(wb)− I1(wb)2)(2g − 1)N.The corollary follows. 5. Optimization : Choosing bIn view of Corollary 1, we are led to the optimization problem of computingmaxb admissible such that I1(wb)<0I1(wb)2I2(wb).In his paper [15], Yu first (his Theorem 1) chooses the functionw(t) =M∑m=0cos (2pi(m+ λ)t)m+ λwhere M is taken equal to 106 and λ = 3/4 (a case for which computations are madeeasier) which gives the bound 1.74246. Yu then proceeds with a numerical optimizationand finally, with λ = 0.75315, he gets the value 1.74217 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 appliedto any truncation of the infinite series (which is non convergent for t = 0) associatedwith Yu’s function. If we go back to the case λ = 3/4, and let M tend to infinity, thisalready gives the bound 1.7424537 . . . , which is the limit of Yu’s function with this choiceof parameter. But, again, one may then move slightly λ. We used a signed continuedfraction method which leads us to consider the value λ = 365/478 (at some step). Withthis choice of λ, we are led to the numerical upper bound 1.7407259 . . . We do not enterinto 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 numericalstudy on functions of the formw(t) = cos((y0 + pi)t)+M∑j=1cjjcos((yj + (2j + 1)pi)t).SIDON 9We used a Maple program and could go up to M = 400 (that is, 801 variables). Thecomputation took about four days on a shared machine equipped with two Intel Xeon E5-2470v2 processors. Notice that, more than time-consuming, this approach is very space-consuming and in fact limited by space considerations. In the above form, we were lookingfor an optimum where the yj are restricted to belong to (0, pi) and the cj to (0, 1). Itturns out that when we increase the number M of variables, the values yj and cj seem toconverge. Here are the first values that are given by the optimization process (obtainedfor M = 400):c01 = 0.448668493767477, c02 = 0.575146465019734, c03 = 0.634139353767643,c04 = 0.668206769165044, c05 = 0.690373909392123, c06 = 0.705944152178521,c07 = 0.717479053644182, c08 = 0.726366349898625, c09 = 0.733423759607086,c10 = 0.739163465377496, c11 = 0.743922783065952, c12 = 0.747933037687434,c13 = 0.751358473065359, c14 = 0.754318115226197, c15 = 0.756900829824045,c16 = 0.759174482613027, c17 = 0.761190238930946, c18 = 0.762988657959701,c19 = 0.764605831570057, c20 = 0.766063873483719, c21 = 0.767398988945215,c22 = 0.768616037123302, c23 = 0.769721510942451, c24 = 0.770739989883381,c25 = 0.771678878036841, c26 = 0.772543457251216, c27 = 0.773353319988327,c28 = 0.774096401927810, c29 = 0.774802358105814, c30 = 0.775461565599078,c31 = 0.776070438424819, c32 = 0.776640535845029, c33 = 0.777213408942223,c34 = 0.777688024987857, c35 = 0.778162522583045, c36 = 0.778618081806088,c37 = 0.779075729278605, c38 = 0.779444959637105, c39 = 0.779857433648994,c40 = 0.780247031029276, c41 = 0.780579370448116, c42 = 0.780921813816887,c43 = 0.781221129831046, c44 = 0.781554783493105, c45 = 0.781870431056320,c46 = 0.782110198962599, c47 = 0.782361619824327, c48 = 0.782643557927602,c49 = 0.782885035586508, c50 = 0.783100192717692,andy00 = 1.69023069423400, y01 = 1.62455004938005, y02 = 1.60400691427448,y03 = 1.59374507362384, y04 = 1.58739065526372, y05 = 1.58292851285127,y06 = 1.57952428074446, y07 = 1.57677070519547, y08 = 1.57444556939989,y09 = 1.57241834643895, y10 = 1.57060466311460, y11 = 1.56895032673690,y12 = 1.56741998541706, y13 = 1.56598348343700, y14 = 1.56462349022195,y15 = 1.56332531960606, y16 = 1.56207725583259, y17 = 1.56086642851363,y18 = 1.55969722035216, y19 = 1.55855788286864, y20 = 1.55745188656436,y21 = 1.55638570641239, y22 = 1.55528462446397, y23 = 1.55421905033814,y24 = 1.55318764446397, y25 = 1.55213468181519, y26 = 1.55113576643217,10 LAURENT HABSIEGER AND ALAIN PLAGNEy27 = 1.55011416521470, y28 = 1.54911054412942, y29 = 1.54815575459570,y30 = 1.54715785448177, y31 = 1.54615472793709, y32 = 1.54518383791521,y33 = 1.54424768835177, y34 = 1.54324227742403, y35 = 1.54234694571695,y36 = 1.54139048590958, y37 = 1.54036349157331, y38 = 1.53942606099970,y39 = 1.53850611740410, y40 = 1.53758211330524, y41 = 1.53663231603874,y42 = 1.53567473396147, y43 = 1.53474740944525, y44 = 1.53383628504159,y45 = 1.53290791051452, y46 = 1.53193597506582, y47 = 1.53097247735348,y48 = 1.53007947174410, y49 = 1.52921326776155, y50 = 1.52829122078524.The interested reader can refer to the complete numerical results available in [8]. Noticethat this function remains close to Yu’s function, which after renormalization can be takenequal tow(t) =M∑m=0λm+ λcos (((2m+ 1)pi + (2λ− 1)pi)t) .Indeed the coefficients cj remains around 0.75 while the coefficients yj are slightly above1.5.Finally considering these values (and those for bigger indices) for M = 400 led us to thevalue 1.74046270371931700 and thus to Theorem 1.Heuristically, it seems that the method could be pushed up to proving the bound 1.74.However, if true and provable by the present method, this could require to use a value ofM much larger than 400.References[1] R. C. Bose, An affine analogue of Singer’s theorem, J. Indian Math. Soc. (N.S.) 6 (1942), 1–15.[2] S. Chowla, Solution of a problem of Erdo˝s and Tura´n in additive number theory, Proc. Nat. Acad.Sci. India. Sect. A. 14 (1944), 1–2.[3] J. Cilleruelo, I. Z. Ruzsa and C. Trujillo, Upper and lower bounds for finite Bh[g] sequences, J. NumberTheory 97 (2002), 26–34.[4] P. Erdo˝s and P. Tura´n, On a problem of Sidon in additive number theory and on some relatedproblems, J. London Math. Soc. 16 (1941), 212–215.[5] B. Green, The number of squares and Bh[g] sets, Acta Arith. 100 (2001), 365–390.[6] L. Habsieger, On finite additive 2-bases, Trans. Amer. Math. Soc. 366 (2014), 6629–6646.[7] L. Habsieger and A. Plagne, Ensembles B2[2] : l’e´tau se resserre, Integers 2 (2002), A2.[8] L. Habsieger and A. Plagne, http://www.cmls.polytechnique.fr/perso/plagne/B2g-num (2016)[9] G. Martin and K. OBryant, Constructions of Generalized Sidon Sets, J. Combin. Theory Ser. A 113(2006), 591–607.[10] G. Martin and K. O’Bryant, The supremum of autoconvolutions, with applications to additive numbertheory, Illinois Journal of Mathematics 53 (2010), No. 1, 219-236.[11] A. Plagne, Recent progress on Bh[g] sets, Congr. Num. 153 (2001), 49–64.[12] S. Sidon, Ein Satz u¨ber trigonometrische Polynome und seine Anwendungen in der Theorie der Fourier-Reihen, Math. Annalen 106 (1932), 536–539.[13] J. Singer, A theorem in finite projective geometry and some applications to number theory, Trans.Amer. Math. Soc. 43 (1938), 377–385.SIDON 11[14] Yu, An upper bound for B2[g] sets, J. Number Theory 122 (2007), 211–220.[15] Yu, A note on B2[g] sets, Integers 8 (2008), A58.E-mail address: habsieger@math.univ-lyon1.frUniversite´ de Lyon, CNRS UMR 5208, Universite´ Claude Bernard Lyon 1, InstitutCamille Jordan, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, FranceE-mail address: plagne@math.polytechnique.frCentre de Mathe´matiques Laurent Schwartz, E´cole polytechnique, CNRS, Universite´Paris-Saclay, 91128 Palaiseau Cedex, France

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

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