A NEW UPPER BOUND FOR FINITE ADDITIVE BASES

Transcription

1 A NEW UPPER BOUND FOR FINITE ADDITIVE BASES C SİNAN GÜNTÜRK AND MELVYN B NATHANSON Abstract Let n, k denote the largest integer n for which there exists a set A of k nonnegative integers such that the sumset A contains {,,,, n } A classical problem in additive number theory is to find an upper bound for n, k In this paper it is proved that lim sup k n, k/k 4789 An extremal problem for finite bases Let N Z denote the nonnegative integers integers, respectively, let A denote the cardinality of the set A Let A be a set of integers, consider the sumset A = {a + a : a, a A} Let S be a set of integers The set A is a basis of order for S if S A The set A is called a basis of order for n if the sumset A contains the first n nonnegative integers, that is, if A is a basis of order for the interval of integers [, n ] := {,,,, n } We define n, A as the largest integer n such that A is a basis of order for n, that is, n, A = max{n : [, n ] A} Rohrbach [6] introduced the extremal problem of determining the largest integer n for which there exists a set A consisting of at most k nonnegative integers such that A is a basis of order for n Let n, k = max{n, A : A N A = k} Rohrbach s problem is to compute or estimate the extremal function n, k The set A is called an extremal k-basis of order if A k n, A = n, k For example, n, = n, = 3 The unique extremal -basis of order is {}, the unique extremal -basis of order is {, } For k = 3 we have n, 3 = 5, the extremal 3-bases of order are {,, } {,, 3} If k A is an extremal k-basis of order, then, A If A is a finite set of k nonnegative integers n, A = n, then n A If a A a > n, then the set A = A \ {a} {n} has cardinality k, n, A n + > n, A Therefore, if A is an extremal k-basis of order n, k = n, then {, } A {,,,, n } A Mathematics Subject Classification Primary B3 Key words phrases Additive bases, segment bases, sumsets The work of SG was supported in part by NSF grant DMS 97 The work of MBN was supported in part by grants from the NSA Mathematical Sciences Program the PSC-CUNY Research Award Program

2 C SİNAN GÜNTÜRK AND MELVYN B NATHANSON If A is an extremal k-basis for n, then A = k A {,,,, n } Rohrbach determined order of magnitude of n, k He observed that if A is a set of cardinality k, then there are exactly k+ ordered pairs of the form a, a with a, a A a a This gives the upper bound k + n, k = k + Ok To derive a lower bound, he set r = [k/] constructed the set We have {,,, r } A Then so Thus, A = {,,,, r, r, r, 3r,, r r} n, A r + A = r k k 4 n, k k 4 + Ok lim inf n n, k k 4 = 5 + = k 4 + Ok n, k lim sup n k = 5 It is an open problem to compute these upper lower limits Mrose [5, ] proved that lim inf n n, k k 7 = 857, this is still the best lower bound Rohrbach used a combinatorial argument to get the nontrivial upper bound lim sup n n, k k 499 Moser [3] introduced a Fourier series argument to obtain lim sup n n, k k 493, subsequent improvements by Moser, Pounder, Riddell [4] produced lim sup n n, k k 4847 Combining Moser s analytic method Rohrbach s combinatorial technique, Klotz [] proved that n, k lim sup n k 48 In this paper, we use Fourier series for functions of two variables to obtain lim sup n n, k k 4789

3 A NEW UPPER BOUND FOR FINITE ADDITIVE BASES 3 We note that Rohrbach used a slightly different function n, k: He defined n, k as the largest integer n for which there exists a set A consisting of k + nonnegative integers such that the sumset A contains the first n + nonnegative integers Of course, Rohrbach s function our function have the same asymptotics Moser s application of Fourier series In this section we describe Moser s use of harmonic analysis to obtain an upper bound for n, k Let A be an extremal k-basis of order Let r,a j denote the number of representations of j as a sum of two elements of A, that is, r,a j = card {a, a A A : a + a = j a a } We introduce the generating function f A q = q a Then f A q + f A q k = f A = A = r,a jq j j A If [, n ] A, then r,a j for all j n Hence there exist integers δj such that where Let f A q + f A q Then q for q, = + q + q + + q n + δjq j, j A { r,a j if j {,,, n }, δj = r,a j otherwise q = δjq j j A f A q + f A q = + q + q + + q n + q Evaluating the generating function identity at q =, we obtain Since, we have k + k = n + n k + Ok The strategy is to find a lower bound for of the form ck + Ok

4 4 C SİNAN GÜNTÜRK AND MELVYN B NATHANSON for some c >, deduce n c k + Ok One obtains a simple combinatorial lower bound for by noting that if a, a A n/ a a, then a + a n Let l denote the number of elements a A such that a n/ Then 3 j n Let δj = j n r,a j ω = e πi/n ll + l be a primitive nth root of unity Let r be an integer not divisible by n Then + ω r + ω r + + ω n r = so f A ω r + f A ω r = + ω r + ω r + + ω n r + j Applying the triangle inequality, we obtain Let Then ω r = fa ω r + f A ω r M = max{ f A ω r : r 4 M k δjω jr = ω r f Aω r k mod n} 5 M k It is also possible to obtain an analytic lower bound for For all integers r not divisible by n, we have M f A ω r = e πira/n = cosπra/n + i sinπra/n, so cosπra/n M sinπra/n M Let ϕt be a function with period with a Fourier series ϕt = a r cosπrt + b r sinπrt r= r=

5 A NEW UPPER BOUND FOR FINITE ADDITIVE BASES 5 whose Fourier coefficients converge absolutely, that is, a r + b r < Define C = Cn by r= C = r= a r For any integer a we have a ϕ = n a r cosπra/n + b r sinπra/n r= r= = a r cosπra/n + b r sinπra/n so r= r= n r r= n r r= r= n r = a r cosπra/n + b r sinπra/n + k a r, a ϕ M n a r + b r + kc r= n r Let α α be real numbers such that ϕt α for t < / ϕt α for / t < Recall that l denotes the number of elements a A such that n/ a n Then a ϕ k lα + lα = kα α α l n We obtain the inequality 6 kα α α l M a r + b r + kc In this way, the function ϕt produces a lower bound for M, which, by 5, gives a lower bound for Moser applied inequality 6 to the function r= n r ϕt = cos4πt + sinπt, whose nonzero Fourier coefficients are a = / b = Then C = for n 3, a ϕ 3M n r= n r

6 6 C SİNAN GÜNTÜRK AND MELVYN B NATHANSON The function ϕt satisfies the inequality { ϕt for t < / 3 for / t <, so This implies that a ϕ n k l 3l = k 4l M 3 a ϕ n k 4l 3 Defining λ = l/k µ = M/k, which both lie in [, ], we obtain the constraint 4 3 λ + µ 3 Recalling the combinatorial lower bound 3 the analytical bound 5, we next obtain k max λ, µ maxλ, µ k k It is now easy to see that hence we obtain maxλ, µ 4 3 λ + µ , k 98 k Inserting this into inequality, we obtain so n k + k = n + n + k 98 k, k + k 4898k + k, 98 which, in fact, has a slightly better constant than derived by Moser originally The constant in this estimate can be further improved by optimizing the function ϕ In the next section we shall employ a more general method using Fourier series in two variables that ultimately yields an even better lower bound for 3 Fourier series in two variables We use the same notation as in the previous section In particular, l denotes the number of integers a A such that a n/ Let L denote the number of pairs a, a A A such that a + a n Then L l, k L is the number of pairs a, a A A such that a + a n We have the combinatorial lower bound 7 r,a j = L + l L j n

7 A NEW UPPER BOUND FOR FINITE ADDITIVE BASES 7 Let ϕt, t be a function with period in each variable with a Fourier series ϕt, t = ˆϕr, r e πirt e πirt r Z r Z whose Fourier coefficients converge absolutely, that is, ˆϕr, r < r Z r Z We choose ϕt, t with zero mean, that is, Let let ˆϕ, = ϕt, t dt dt = R = {t, t [, [, : t + t < } R = {t, t [, [, : t + t } If a, a A a + a n, then a /n, a /n R If a + a n, then a /n, a /n R Let α α be real numbers such that ϕt, t α for t, t R ϕt, t α for t, t R We choose the function ϕt, t such that Then 8 a A a A α > α a ϕ n, a k Lα + Lα = α k α α L n We can rewrite this sum as follows: a ϕ n, a = ˆϕr, r e πira/n e πira/n n a A a A a A a A r Z r Z = ˆϕr, r e πira/n r Z r Z a A a A = ˆϕr, r f A ω r f A ω r r Z r Z Consider the partition of the integer lattice Z = S S S : S ={r, r Z : r r mod n} S ={r, r Z : r mod n, r mod n} {r, r Z : r mod n, r mod n} S ={r, r Z : r mod n, r mod n} We define C i = C i n for i =,, by C i = ˆϕr, r r,r S i e πira/n

8 8 C SİNAN GÜNTÜRK AND MELVYN B NATHANSON Recall that f A ω r M if r is not divisible by n f A ω r = k if r is divisible by n Then a 9 ϕ n, a C k + C km + C M n a A a A Combining inequalities 8 9, we obtain Since α > α, we have α k α α L C k + C km + C M We again define L α C k C km C M α α Since M k, we have µ = M k µ By inequality 7, we have L, so k L k α C C µ C µ α α By inequality 5, we also have M k, so k max µ, α C C µ C µ α α k Since the series of Fourier coefficients of ϕt, t converges absolutely since ˆϕ, =, we can arrange the Fourier series in the form of a sum over concentric squares ˆϕr, r e πirt e πirt R= max r, r =R For any ε > there exists an integer N = Nε such that n=n max r, r =n ˆϕr, r < ɛα α For all n N, we shall approximate the sums C, C, C by, C axial, C main, respectively, where C axial = r Z r ˆϕ, r + ˆϕr, C main = r Z r ˆϕr, r r Z r

9 A NEW UPPER BOUND FOR FINITE ADDITIVE BASES 9 Then α C C µ C µ α C axialµ C mainµ so = C + C C axial µ + C C main µ C + C C axial + C C main ˆϕr, r max r, r N < ɛα α, α C C µ C µ α C axial µ C main µ < ε α α α α It follows from inequality that k max µ, α C axial µ C main µ ε α α k Let ρ = inf µ max µ, α C axial µ C main µ α α From the definition of ρ in, we now have k ρ ε k Applying identity, we obtain k + k = n + n + ρ εk k Therefore, we obtain the desired estimate ρ + ε n k + k, where the number ρ depends only on the function ϕt, t ɛ > can be arbitrarily small It is clear that we always have ρ, that ρ > if only if α > It is also clear that when α, we have ρ = ξ, where ξ is the unique solution in [, ] to the quadratic equation ie, ξ = α C axial ξ C main ξ α α, α α + C main ξ + C axial ξ α =, which yields the formula C axial + Caxial 3 ρ = + 4α α α + C main α α + C main Hence we have an optimization problem in which we maximize ρ over all real valued functions ϕ defined on the unit square [, such that ϕ has zero mean ϕ > on R We do not know the optimal function for this problem, but we have

10 C SİNAN GÜNTÜRK AND MELVYN B NATHANSON found a simple piecewise polynomial function that improves Klotz s upper bound for n, k Before we proceed to the main result of this paper, which also includes the definition of this function, let us present some of the heuristics which have lead us to our educated guess First, without loss of generality, we may assume that α = We then necessarily have = ϕt, t dt dt + ϕt, t dt dt R R + α, hence α We also have C axial ˆϕr, + ˆϕ, r = ϕ, t + ϕt, dt, r C main ˆϕr, r = r ϕt, tdt In any case we are interested in the positive root ξ κ,τ of the equation κξ + τξ = where κ = α + C main 3 τ = C axial Clearly, the smaller κ τ are, the larger this root will be The bounds κ 3 τ already imply that ξ κ,τ 3, hence ρ = ξ κ,τ 9 In reality, α < because equality can happen only if ϕ is constant on both R R, in which case ˆϕ is not absolutely summable This results in the heuristic that if we try to push α close to, then C axial C main will become large, conversely if we try to push C axial C main close to their respective minimum values, then ϕ may not be bounded from below on R by a small value The right trade-off between these two competing quantities will result in the solution of this optimization problem It is interesting to note that the value of ρ is fairly robust with respect to variations in κ τ, which we will only be able to estimate numerically not compute exactly The following lemma gives an explicit estimate for this purpose: Lemma Let ξ κ,τ ξ κ,τ be the respective positive roots of the equations κξ + τξ = κ ξ + τ ξ = Let ρ = ξκ,τ ρ = ξκ,τ If minκ, κ 3 minτ, τ, then 4 ρ ρ 54 κ κ + 8 τ τ The proof of this lemma is given in the Appendix Now we can state prove the main theorem of this paper Theorem lim sup n n, k k 4789 Proof We define the function ϕt, t on the unit square [, by {, t, t 5 ϕt, t = R 4 t t t t 6, t, t R The graph of this function is plotted in Figure

11 A NEW UPPER BOUND FOR FINITE ADDITIVE BASES Figure Graph of the function ϕ defined in 5 Clearly we have α = Computation of the three other parameters used in formula for ρ yields α = 5 = 3747, 5/3 978 C axial 989, C main Taking κ = = 94867, τ = 989, we obtain ρ > 44, κ κ < τ τ <, so that ρ ρ < Hence ρ ρ ρ ρ > 4 Consequently choosing ɛ sufficiently small using, we obtain n 4789 k + k The details of the computations are in the Appendix to this paper This completes the proof 4 Open problems A major open problem concerning the extremal function n, k = max{n, A : A N A k}

12 C SİNAN GÜNTÜRK AND MELVYN B NATHANSON is to compute lim inf n n, k/k lim sup n n, k/k, to determine if the limit n, k lim n k exists We have no conjecture about the existence of this limit, nor about the values of the lim inf lim sup It is also difficult to compute the exact values of the function n, k We can generalize the extremal functions n, A n, k as follows Let A be a finite set of integers, let m, A denote the largest integer n such that the sumset A contains n consecutive integers Let m, k = max{m, A : A Z A k} Let l, A denote the largest integer n such that the sumset A contains an arithmetic progression of length n, let l, k = max{l, A : A Z A k} We can also define the extremal function Then so n, k = max{n, A : A Z A k} n, A n, A m, A l, A, n, k n, k m, k l, k For any integer t set A, we have the translation A + t = {a + t : a A} The functions l m are translation invariant, that is, l, A + t = l, A m, A + t = m, A We also have the trivial upper bound l, k k+, but it is an open problem to obtain nontrivial upper bounds for any of the extremal functions n, k, m, k, or l, k We describe here the computations Appendix Proof of Lemma We start with the formula ξ κ,τ = τ + τ + 4κ κ = τ + τ + 4κ We next evaluate the partial derivatives of ξ κ,τ with respect to κ τ: 6 7 ξ κ,τ κ ξ κ,τ τ 4 = τ + 4κ τ + τ + 4κ, = τ + 4κ τ + τ + 4κ from which it follows that in the set {κ, τ : κ 3, τ }, we have ξκ,τ τ These bounds then imply ξ κ,τ ξ κ,τ 36 κ κ + τ τ ξκ,τ κ 36

13 A NEW UPPER BOUND FOR FINITE ADDITIVE BASES 3 We then note that ξ κ,τ 3, which yields ρ ρ = ξ κ,τ ξ κ,τ ξ κ,τ + ξ κ,τ 3 ξ κ,τ ξ κ,τ, hence the result of the lemma Absolute summability of ˆϕ While the result explained in this subsection is elementary, we will provide a certain amount of detail in its derivation because our main concern is more than absolute summability of ˆϕ We would like to provide explicit estimates on the rate of the convergence; this will be necessary later in the section when we will analyze the accuracy of the numerical computation of the constants C main C axial Lemma Let f be a smooth function on [, ] Then for all L n, the following formula holds: 8 ˆfn = L k= f k f k f πin k+ + L+ n πin L+ Proof The case L = follows from integration by parts the general case follows from iterating this result Theorem Let F be a smooth function on R which vanishes on the boundary of R, ie, Define F t, t = F, t = F t, =, for all t [, ] Ψ F t, t = {, if t, t R, F t, t, if t, t R Then the Fourier series expansion of Ψ F is absolutely convergent Note: Later we will simply set ϕ = Ψ F + Proof We will prove this result by deriving a suitable decay estimate on ˆΨ F r, r, where { } ˆΨ F r, r = e πirt F t, t e πirt dt dt t The case r = or r = Due to the symmetry on the assumptions on F, it suffices to consider only one of these cases Let us assume that r = Define so that J t = t F t, t dt 9 ˆΨF r, = Ĵr Clearly we have J = J = Setting L = f = J in Lemma, we see that for r ˆΨ F r, πr J J + J = O r

14 4 C SİNAN GÜNTÜRK AND MELVYN B NATHANSON With a similar estimate for ˆΨ F, r, we have ˆΨ F r, + ˆΨ F, r < r The case r r We will derive a general formula for ˆΨ F r, r To do this, we momentarily forget that F vanishes on the boundary of R, for t [, ], define the following functions: g t = F t, t, h t = F t,, g t = F t, t, h t = F, t, g t = F t, t, h t = F t,, g 3 t = F t, t, h 3t = F, t We start with the formula for ˆΨ F r, r above Integrating by parts in the second variable, we obtain { [ e πir t ] t= ˆΨ F r, r = dt e πirt F t, t πir t = t } e πirt dt F t, t t πir = ĝ r r πir ĥr + ˆΨ F r, r We apply the same method to ˆΨ F r, r, but integrate by parts in the first variable This results in ˆΨ F r, r = ĝ r r πir ĥr + πi ĝ r r r r ĥr + ˆΨ F r, r We repeat the first two steps in the same order, which gives us ˆΨ F r, r = ĝ r r πir ĥr + πi ĝ r r r r ĥr + πi 3 r r ĝ r r ĥr + πi 4 r ĝ 3 r r ĥ3r + ˆΨ r F r, r Note that from our assumptions on F, we have g = h = h = We will have two subcases: r = r = r In this case, we easily see from the second formula above that ˆΨ F r, r ĝ + Ψ F 4π r r r This case is slightly more subtle We first note that g = h = It is also true that g = F, To see this, note that F, = F,, =, both of which follow from the

15 A NEW UPPER BOUND FOR FINITE ADDITIVE BASES 5 fact that F vanishes on the boundary of R The function g being smooth otherwise, we conclude that ĝ r r g g + g 4π r r The estimates for g h are simpler in nature We use the bounds as well as ĝ r r g g + g, π r r ĥr h h + h, π r ĝ 3 r r g 3, ĥ3r h 3, ˆΨ F r, r Ψ F Putting all these together, we see that ˆΨ F r, r = O r r r r + r r r r + r r which is easily verified to be summable over all admissible values of r r We will return to this shortly for a more explicit estimate Explicit numerical estimates In this subsection we will work with the specific function ϕ in 5 for which F t, t = 4 t t t t 6 We will numerically estimate C axial C main using appropriate tail bounds on their defining infinite series This procedure is illustrated in Figure Estimating the value of C axial Since F is symmetric, we have ˆϕr, = ˆϕ, r We use the formula 9 to evaluate ˆϕr, It is a simple calculation to show that J t = 5 t t t t 9 Using this expression, we find that J J = 5 J Hence by, we obtain the estimate If we define J / = 8 5 ˆϕr, = ˆϕ, r π r 3 C axial N = ˆϕr, + ˆϕ, r, r N

16 6 C SİNAN GÜNTÜRK AND MELVYN B NATHANSON r r Figure Estimating the values of C axial C main Circled dots correspond to the Fourier coefficients along the diagonal for which a separate decay estimate holds then it follows that C axial C axial N π N < 5 N To estimate C axial N, we still need the actual expression for ˆϕr,, which is given in 3 Taking N = 5, numerical computation shows that C axial N = 978 ; hence it follows that 978 C axial 989 Estimating the value of C main We shall estimate the diagonal terms first We have from which we obtain g t = 4t t ĝ = 4, F t, t = t t t t t t 6 4, from which we obtain t t t t t t Ψ F L [, 8

17 A NEW UPPER BOUND FOR FINITE ADDITIVE BASES 7 Hence by, we obtain the estimate from which it follows that r =N+ ˆϕr, r 3 π r ˆϕr, r 6 π N We next estimate ˆϕr, r in the case when r r We begin by noting that We have ĝ r r = g t = 4 + 5t t, h t = t 6, g 3 t = 4 5t + t, h 3 t = 68 t 5, π r r, r r F t, t = 44 t t 5 + t + t 5t 5t + 3t t, from which we obtain ĝ r r = 6 π r r, r r, ĥr 8 π r, r, ĝ 3 r r 38, ĥ3r 8, Ψ F = 8 Putting these together, we finally obtain the estimate 4 ˆϕr, r 5 π 4 r r r r + 4 π 4 The following is a simple lemma: Lemma 3 For any N, one has 5 6 R=N+ R=N+ max r, r =R min r, r r r max r, r =R min r, r r r Proof The first inequality simply follows from max r, r =R min r, r < 4π 3 N, r r π r r r r < N r r = 8 R R r= r 4π 3 R

18 8 C SİNAN GÜNTÜRK AND MELVYN B NATHANSON For the second inequality, we first use the symmetries to write max r, r =R min r, r r r Using the identity r r r r = 4 R R r= Cauchy-Schwarz inequality we have R R 4 R r= rr r + 4 R R r= rr r = RrR r + RR r, rr r = 4 R r= R rr r + r= For the remaining terms, we use the trivial estimate Hence the result follows If we define 4 R max r, r =R min r, r r r R r= 7 C main N = then we have rr + r R 4 < 4 R rr + r R 4 R r < 4π 3 π r r r r < R, N R= 8 C main C main N < max r, r =R min r, r ˆϕr, r, 34 π + 4 π 4 N < 4 N R For N = 4, numerical computation using the formulas 3 3 reveals that C main N = ; hence with the above error estimate, we have C main Explicit expressions for ˆϕr, r The following formulas have been computed using Mathematica, though it is also possible to compute them easily using the iterative procedure based on integration by parts which was outlined in this section earlier 3 ˆϕr, = 5 4π r 6 π r + 45 π 4 r 4 35 π 6 r 6 i 6 7π 3 r π r 35 8π 4 r π 6 r 6

19 A NEW UPPER BOUND FOR FINITE ADDITIVE BASES 9 3 ˆϕr, r = π r π r + 35 π 4 r π 6 r 6 + i 55 π 3 r 3 6 π r + 63 π 4 r π 6 r 6 3 ˆϕr, s = π 8 r 6 r s π 6 r 4 r s 35 π 4 r r s π 8 r s s π 8 r r s s 5 4 π 6 r s s π 8 r r s s π 8 r 3 r s s 3 75 π 6 r r s s 35 3 π 4 r s s + 5 π 8 r 4 r s s 75 π 6 r r s s 5 + π 8 r 5 r s s 75 π 6 r 3 r s s + 5 π 4 r r s s 575 i 4 π 9 r 7 r s + 55 π 7 r 5 r s 5 π 5 r 3 r s π 9 r s s π 9 r r s s π 7 r s s π 9 r r s s π 9 r 3 r s s 4 75 π 7 r r s s 5 4 π 5 r s s π 9 r 4 r s s 75 3 π 7 r r s s π 9 r 5 r s s 75 π 7 r 3 r s s + 5 π 5 r r s s + 5 π 9 r 6 r s s References 75 π 7 r 4 r s s + 5 π 5 r r s s [] Gerd Hofmeister, Thin bases of order two, J Number Theory 86, no, 8 3 [] Walter Klotz, Eine obere Schranke für die Reichweite einer Extremalbasis zweiter Ordnung, J Reine Angew Math , 6 68 [3] L Moser, On the representation of,,, n by sums, Acta Arith 6 96, 3 [4] L Moser, J R Pounder, J Riddell, On the cardinality of h-basis for n, J London Math Soc , [5] A Mrose, Untere Schranken für die Reichweiten von Extremalbasen fester Ordnung, Abh Math Sem Univ Hamburg , 8 4 [6] H Rohrbach, Ein Beitrag zur additiven Zahlentheorie, Math Zeit 4 937, 3 Courant Institute, New York University, New York, New York address: gunturk@courantnyuedu Department of Mathematics, Lehman College CUNY, Bronx, New York 468 address: melvynnathanson@lehmancunyedu