Dual linear programming bounds for sphere packing via modular forms
DDUAL LINEAR PROGRAMMING BOUNDSFOR SPHERE PACKING VIA MODULAR FORMS
HENRY COHN AND NICHOLAS TRIANTAFILLOU
Abstract.
We obtain new restrictions on the linear programming bound forsphere packing, by optimizing over spaces of modular forms to produce feasiblepoints in the dual linear program. In contrast to the situation in dimensions 8and 24, where the linear programming bound is sharp, we show that it comesnowhere near the best packing densities known in dimensions 12, 16, 20, 28, and32. More generally, we provide a systematic technique for proving separationsof this sort. Introduction
The sphere packing problem asks for the densest packing of congruent spheres in R d . In other words, what is the greatest proportion of R d that can be covered bycongruent balls with disjoint interiors? The case d = 1 is trivial, d = 2 was solved byThue [29], and d = 3 was solved by Hales [17] with a computer-assisted proof thathas since been formally verified [18]. These proofs make essential use of the geometryof packings in R d in a way that seems difficult to extend to higher dimensions,and so another approach is needed when d is large. Based on a long history oflinear programming bounds in coding theory, Cohn and Elkies [6] developed a linearprogramming bound for sphere packing. It yields the best upper bounds known forthe packing density in high dimensions [12], and Cohn and Elkies conjectured thatthe linear programming bound is sharp when d = 8 or d = 24.In a recent breakthrough, Viazovska [31] proved this conjecture for d = 8, andthus showed that the E root lattice yields the densest sphere packing in R .Shortly thereafter, Cohn, Kumar, Miller, Radchenko, and Viazovska [10] provedthe conjecture for d = 24. These are the only two cases beyond d = 3 in which thesphere packing problem has been solved.These advances raise numerous questions. Is it possible that the linear program-ming bound is sharp in some other dimensions? Could it even be sharp in everydimension? (Surely not, but why not?) What happens in R , and why does thatcase seemingly not behave like R and R ? These questions remain mysterious,but in this paper we take some initial steps towards answering them.The difficulty in analyzing the linear programming bound stems from the use ofan auxiliary function, which must satisfy certain inequalities. The quality of thebound depends on the choice of this function, and optimizing the bound amountsto optimizing a functional over the infinite-dimensional space of auxiliary functions.This optimization problem has not been solved exactly except when d ∈ { , , } . Date : September 10, 2019.Triantafillou was partially supported by an internship at Microsoft Research New England, aNational Science Foundation Graduate Research Fellowship under grant a r X i v : . [ m a t h . M G ] S e p HENRY COHN AND NICHOLAS TRIANTAFILLOU − − − − − Dimension
Sph e r e p a c k i n g d e n s i t y Linear programming boundDensest known packingDual (lower) bound
Figure 1.1.
The upper curve is the linear programming boundcomputed using the best auxiliary functions currently known, whilethe white circles are the densest sphere packings currently known(see [13, pp. xix–xx]). Our new obstructions, drawn as black circles,are lower bounds for the linear programming bound. They showthat further optimizing the choice of auxiliary function cannotimprove the linear programming bound by much.In other dimensions, we can approximate the true optimum by using a computer tooptimize over a finite-dimensional subspace. The resulting auxiliary function alwaysproves some bound for the sphere packing density, and we expect it to be close tothe optimal linear programming bound if the subspace is large and generic enough.However, nobody has been able to determine how close it must be. What if thesenumerical computations are woefully far from the true optimum? If that were thecase, then they would shed very little light on the linear programming bound. Itis even possible, albeit implausible, that the linear programming bound might besharp for relatively small values of d that nobody has noticed yet.As shown in Figure 1.1, the linear programming bound seems to vary smoothly asa function of dimension, and the sharp bounds in 8 and 24 dimensions fit perfectlywith the curve as a whole. These observations raise our confidence that the numericaloptimization is not in fact misleading. However, there remains a fundamental gapin the theory of the linear programming bound: how can one prove a correspondinglower bound, beyond which no auxiliary function can pass? In optimization terms,such a bound amounts to a dual linear programming bound , which controls howgood the optimal linear programming bound could be.In this paper, we show how to compute such a bound when the dimension isa multiple of four, by optimizing over spaces of modular forms. (We expect thatother dimensions work similarly, but we have not carried out the modular formcalculations in those cases.) Our results for dimensions 12, 16, 20, 28, and 32 are UAL LINEAR PROGRAMMING BOUNDS FOR SPHERE PACKING. . . 3 shown in Figure 1.1 and Table 6.1. The most noteworthy cases are dimensions 12and 16, where the Coxeter-Todd and Barnes-Wall lattices are widely conjectured tobe optimal sphere packings:
Theorem 1.1.
The linear programming bound for the sphere packing density in R is greater than . times the density of the Barnes-Wall lattice, and the bound in R is greater than . times the density of the Coxeter-Todd lattice. In particular,the linear programming bound cannot prove that either lattice is an optimal spherepacking. Unsurprisingly, in neither case is the linear programming bound even close toreaching the best density known. The ratios 1 . .
686 are almost certainlynot quite optimal, and we expect that they could be improved to 1 .
712 and 1 . The linear programming bound.
Before proceeding further, let us reviewhow the linear programming bound works. Recall that a sphere packing in R d is adisjoint union (cid:83) x ∈ C B ( x, ρ ) of open unit balls of some fixed radius ρ and centeredat the points of some subset C of R d .Given a sphere packing P , the upper density ∆ P of P is defined by∆ P = lim sup r →∞ vol( B ( x, r ) ∩ P )vol( B ( x, r ))for any x ∈ R d (the upper density does not depend on the choice of x ). If the limitexists, and not just the limit superior, then we say that P has density ∆ P . The sphere packing density in R d is ∆ d = sup P⊂ R d ∆ P , where the supremum is over sphere packings P . We will often renormalize and workwith the upper center density δ P = ∆ P vol( B (0 , r →∞ B ( x, r ) ∩ C )vol( B ( x, r )) · vol( B (0 , ρ ))vol( B (0 , , which measures the number of center points per unit volume in space if we usespheres of radius ρ = 1. Of course the center density has no theoretical advantageover the density, but it is often convenient not have to carry around the factorof vol( B (0 , π d/ / ( d/ δ = 1, while ∆ = π /
12! =0 . . . . .We normalize the Fourier transform (cid:98) f of an integrable function f : R d → R by (cid:98) f ( y ) = (cid:90) R d f ( x ) e − πi (cid:104) x,y (cid:105) dx, HENRY COHN AND NICHOLAS TRIANTAFILLOU − − xf ( x ) − − y (cid:98) f ( y ) Figure 1.2.
A sample auxiliary function and its Fourier transform(namely, f ( x ) = (1 − x ) e − x on R , with r = 1).where (cid:104)· , ·(cid:105) denotes the usual inner product on R d . Cohn and Elkies [6] showed how to use harmonic analysis to bound the sphere packing density as follows: Theorem 1.2 (Cohn and Elkies [6]) . Let f : R d → R be a continuous, integrablefunction, such that (cid:98) f is integrable as well and (cid:98) f is real-valued (i.e., f is even).Suppose f and (cid:98) f satisfy the following inequalities for some positive real number r : (1) f (0) > and (cid:98) f (0) > , (2) f ( x ) ≤ for | x | ≥ r , and (3) (cid:98) f ( y ) ≥ for all y .Then every sphere packing in R d has upper center density at most f (0) (cid:98) f (0) · (cid:16) r (cid:17) d . The linear programming bound in R d is the infimum of the center density upperbound f (0) (cid:98) f (0) · (cid:16) r (cid:17) d over all auxiliary functions f satisfying the hypotheses of Theorem 1.2. See Figure 1.2for an example of an auxiliary function, which is far from optimal.Without loss of generality, we can assume that the auxiliary function f is radial,because we can simply average its rotations about the origin. For a radial function f , we write f ( t ) with t ∈ [0 , ∞ ) to denote the common value f ( x ) with | x | = t . If f is radial, then (cid:98) f is radial as well, and (cid:98) f ( y ) = 2 π | y | d/ − (cid:90) ∞ f ( t ) J d/ − ( t | y | ) t d/ dt, where J d/ − is the Bessel function of the first kind of order d/ − Strictly speaking, the paper [6] imposed stronger hypotheses on f , but one can easily removethose hypotheses by mollifying f , using the approach from the first paragraph of Section 4 in [5].The fact that they could be removed was first observed in [8]. UAL LINEAR PROGRAMMING BOUNDS FOR SPHERE PACKING. . . 5
The density bound f (0) (cid:98) f (0) · (cid:16) r (cid:17) d is invariant under replacing f with x (cid:55)→ f ( ρx ) and r with r/ρ for any scalingfactor ρ ∈ (0 , ∞ ). Without loss of generality we can use this invariance to fix r = 1, and we can assume (cid:98) f (0) = 1 as well. Then the constraints on f fromTheorem 1.2 are linear inequalities, and the density bound is also a linear functionalof f . Thus, optimizing the choice of f amounts to solving an infinite-dimensionallinear optimization problem, which explains the name “linear programming bound.”In practice, however, fixing r may not lead to the prettiest answers. For example,Cohn and Elkies found more elegant behavior if one instead fixes f (0) = (cid:98) f (0) andlets r vary (see Section 7 of [6]).The best choice of f is not known, except when d ∈ { , , } , and little is knownabout how good the optimal bound might be. It is not hard to produce upper boundsby numerically optimizing over finite-dimensional spaces of functions, and in mostcases these upper bounds seem to be close to the optimal linear programming bound(see [7] for the most extensive calculations so far). However, these computationalmethods leave open the possibility that other auxiliary functions might prove muchbetter bounds.What sort of obstructions prevent the linear programming bound from reachingthe density of the best sphere packing? In this paper we provide a partial answer,with an algorithm to compute such obstructions via linear programming over spacesof modular forms of weight d/
2. The algorithm is based on optimizing a summationformula for radial Schwartz functions, which is an analogue of Voronoi summation.The remainder of the paper is organized as follows. In Section 2, we present ageneral framework for computing dual linear programming bounds. We describeour algorithm in Section 3, and we prove the summation formula underlying thealgorithm in Section 4. In Section 5, we expand on the final step of our algorithmby describing a method for checking in finite time that all of the coefficients of the q -expansion of a given modular form are nonnegative. Finally, we present a table ofnew lower bounds in Section 6, and we conclude with open problems in Section 7.2. Duality
Computing a bound for the objective function in a linear program is typicallystraightforward: it just amounts to finding a feasible point in the dual linearprogram. The difficulty in our case is that the optimization problems are infinite-dimensional. The primal problem is relatively tractable, because the auxiliaryfunctions in Theorem 1.2 are well behaved in practice. We can approximate themwith polynomials times Gaussians, and using high-degree polynomials yields excellentresults. For example, in R the resulting center density bounds seem to converge to0 . . . . as the polynomial degree tends to infinity, and we believe this number is theoptimal linear programming bound for 16 dimensions, correct to 32 decimal places.Unfortunately, the dual problem is much less tractable. It amounts to optimizingover a space of measures, and we believe the optimal measures will be singular(specifically, supported on a discrete set of radii). In particular, we know of no HENRY COHN AND NICHOLAS TRIANTAFILLOU simple family of measures we can use to approximate them fruitfully. Instead, thedual problem appears to be quite a bit more subtle.In Section 4 of [5], Cohn formulated the dual linear program as follows. Here, δ denotes a delta function at the origin, and (cid:98) µ is the Fourier transform of µ as atempered distribution. Proposition 2.1.
Let µ be a tempered distribution on R d such that µ = δ + ν with ν ≥ , supp( ν ) ⊆ { x ∈ R d : | x | ≥ r } for some r > , and (cid:98) µ ≥ cδ for some c > .Then the linear programming bound in R d is at least c · (cid:16) r (cid:17) d . Sketch of proof.
Let f : R d → R be an auxiliary function satisfying the hypothesesof Theorem 1.2, where we use scaling invariance to ensure that the same value of r works for both f and µ . If f and (cid:98) f are rapidly decreasing, then the inequalities on f and µ imply that f (0) ≥ (cid:90) R d f µ = (cid:90) R d (cid:98) f (cid:98) µ ≥ c (cid:98) f (0) , and thus f (0) (cid:98) f (0) ≥ c, as desired. More general auxiliary functions must be mollified, as described inSection 4 of [5], after which the same argument applies to them as well. (cid:3) The difficulty in applying this proposition is how to find a plentiful source ofdistributions µ that could satisfy the hypotheses. One source is Poisson summationfor lattices, which says that for any lattice Λ in R d , the Fourier transform of thedistribution (cid:88) x ∈ Λ δ x is 1vol( R d / Λ) (cid:88) y ∈ Λ ∗ δ y , where Λ ∗ is the dual lattice. Thus, the hypotheses of Proposition 2.1 are satisfiedwith c = 1 / vol( R d / Λ) and r = min x ∈ Λ \{ } | x | . The resulting lower bound amountsto proving Theorem 1.2 for lattice packings.In principle, one could try to improve on individual lattices by using a linearcombination of Poisson summation formulas for different lattices (see, for example,the bottom of page 351 in [5]). However, that does not seem fruitful in general.Instead, we use the following analogue of Voronoi summation to produce distributionsfrom modular forms. For definitions related to modular forms, see [16]. In particular,recall that the slash operator is defined as follows: if M = (cid:18) a bc d (cid:19) ∈ GL ( R ) anddet M >
0, then ( f | k M )( z ) := ( ad − bc ) k/ ( cz + d ) − k f (cid:18) az + bcz + d (cid:19) . UAL LINEAR PROGRAMMING BOUNDS FOR SPHERE PACKING. . . 7
Proposition 2.2.
Let d = 2 k with k ∈ N , let g ∈ M k (Γ ( N )) be a modular formof weight k for the congruence subgroup Γ ( N ) , let w N = (cid:18) − N (cid:19) , and let (cid:101) g ( z ) = i k ( g | k w N )( z ) = i k N k/ z k g (cid:18) − N z (cid:19) be i k times the image of g under the full level N Atkin-Lehner operator (so that g = i k (cid:101) g | k w N as well). Let the q -expansions of g and (cid:101) g be g ( z ) = ∞ (cid:88) n =0 a n q n and (cid:101) g ( z ) = ∞ (cid:88) n =0 b n q n , where q = e πiz . Then for every radial Schwartz function f : R d → C , ∞ (cid:88) n =0 a n f ( √ n ) = (cid:18) √ N (cid:19) d/ ∞ (cid:88) n =0 b n (cid:98) f (cid:18) √ n √ N (cid:19) . In particular, if δ r denotes a delta function supported on the sphere of radius r about the origin in R d , then this proposition says that the tempered distributions ∞ (cid:88) n =0 a n δ √ n and (cid:18) √ N (cid:19) d/ ∞ (cid:88) n =0 b n δ √ n/N are Fourier transforms of each other. Our algorithm will optimize over distributionsof this form. The advantage of these distributions is that their supports help enforcethe constraint that supp( ν ) ⊆ { x ∈ R d : | x | ≥ r } in Proposition 2.1.For comparison, the techniques in Section 5 of [7] produce what appear to beclose numerical approximations to the optimal distributions µ . They have the form µ = (cid:88) n ≥ c n δ r n with radii given by 0 = r < r < r < · · · and tending to infinity, coefficients c n >
0, and (cid:101) µ = µ . For example, in R the first few radii and coefficients are listedin Table 2.1. The only drawback is that the results of these calculations are merelyconjectural: we do not know whether such a distribution actually exists.Our approach in this paper amounts to approximating the optimal µ with adistribution µ (cid:48) whose existence follows from Proposition 2.2. For comparison,Table 2.1 shows the best µ (cid:48) we have obtained, which we computed using theparameters N = 96 and T = 20 in the notation of the next section. This distributionis of the form µ (cid:48) = (cid:80) n ≥ c (cid:48) n δ r (cid:48) n , with Fourier transform (cid:98) µ (cid:48) = (cid:80) n ≥ c (cid:48)(cid:48) n δ r (cid:48)(cid:48) n . In thetable, we have rescaled the distribution µ (cid:48) so that c (cid:48) = c (cid:48)(cid:48) = 1. Note that r ≈ r (cid:48) ≈ r (cid:48)(cid:48) ≈ r (cid:48)(cid:48) ,r ≈ r (cid:48) ≈ r (cid:48) ≈ r (cid:48)(cid:48) ≈ r (cid:48)(cid:48) , and r ≈ r (cid:48) ≈ r (cid:48) ≈ r (cid:48)(cid:48) ≈ r (cid:48)(cid:48) , and the sums of the corresponding coefficients are also near each other. Theapproximation to µ is not yet very close, but one can already see µ roughly emergingfrom µ (cid:48) . HENRY COHN AND NICHOLAS TRIANTAFILLOU
Table 2.1.
Radii and coefficients for dual distributions in R . n r n c n . . . . . . . . . . . . . . . . . . . . . . . .n r (cid:48) n c (cid:48) n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .n r (cid:48)(cid:48) n c (cid:48)(cid:48) n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An algorithm for dual linear programming bounds
Proposition 2.2 allows modular forms for the congruence subgroup Γ ( N ), butfor simplicity we will restrict our attention to those for the larger group Γ ( N )(equivalently, to modular forms for Γ ( N ) that have trivial Nebentypus). There issome loss of generality, but this case serves as an attractive proving ground for thegeneral theory, and it should suffice when the dimension d is a multiple of 4.Specifically, let k = d/ M k (Γ ( N )) be the space ofmodular forms of weight k for Γ ( N ). Recall that this space has a basis consistingof modular forms with rational coefficients in their q -expansions (see, for example,Corollary 12.3.12 in [15]). Furthermore, the Atkin-Lehner involution on M k (Γ ( N ))preserves the property of having rational coefficients (see Lemma 3.5.3 in [24]).In practice, to simplify Section 5 we also assume that N is not divisible by 16 ,9 , or p for any prime p >
3, but this assumption is not essential.We would like to find a modular form g = (cid:80) n ≥ a n q n in M k (Γ ( N )) with thefollowing properties for some T , where we set (cid:101) g = i k g | k w N = (cid:80) n ≥ b n q n :(1) a = 1 and b > a n ≥ b n ≥ n ≥
0, and(3) a n = 0 for 1 ≤ n < T .Then we use the distribution µ = (cid:88) n ≥ a n δ √ n UAL LINEAR PROGRAMMING BOUNDS FOR SPHERE PACKING. . . 9 in Proposition 2.1. By Proposition 2.2, we have c = (2 / √ N ) d/ b and r = √ T inthe notation of Proposition 2.1. Thus, we obtain a lower bound of b (cid:18) √ N (cid:19) d/ (cid:32) √ T (cid:33) d for the linear programming bound in R d , and we wish to choose g so as to maximizethis bound. We will do so by linear programming, with one caveat: all our calcula-tions will consider only the terms up to q M in the q -series for some fixed M , and atthe end we must check that the inequalities are not violated beyond that point.Let g , . . . , g dim M k (Γ ( N )) be a basis of M k (Γ ( N )) with rational q -series coeffi-cients, and let (cid:101) g j = i k g j | k w N be i k times the image of g j under the full level N Atkin-Lehner involution. We write the q -expansions of the modular forms g j and (cid:101) g j as g j = ∞ (cid:88) n =0 a jn q n and (cid:101) g j = ∞ (cid:88) n =0 b jn q n , and we fix integers T and M with 1 ≤ T < dim M k (Γ ( N )) < M . These bases and q -series can all be computed algorithmically (see, for example, [28]).Now we write g = (cid:80) j x j g j with respect to our basis, and we optimize over thechoice of coefficients x j by solving the following linear program:maximize (cid:80) j x j b j subject to 1 = (cid:80) j x j a j , (cid:80) j x j a jn for 1 ≤ n < T ,0 ≤ (cid:80) j x j a jn for T ≤ n ≤ M , and0 ≤ (cid:80) j x j b jn for 1 ≤ n ≤ M .These inequalities encode all the desired properties of f and g , except that weexamine only the terms up to q M in the q -series.We hope that if M is large enough, then all the terms beyond q M will havenonnegative coefficients automatically, and we attempt to use asymptotic boundsto confirm that all of the coefficients of g and (cid:101) g are nonnegative (see Section 5). Ifthis verification fails, we can increase M and attempt the optimization problemagain. In practice, M = 2 · dim M k (Γ ( N )) typically seems to be sufficient for thealgorithm to succeed, and it works for all the numerical results we report in thispaper.To find the best possible bounds, we run the method for several values of N and T . Larger values of N typically yield better results, but not always. It seems difficultto predict the best values for T in general, although they also tend to increase as N increases. See Section 6 for the results of this method applied to the spaces M k (Γ ( N )) of modular forms of weight k ∈ { , , , , } and level N = 24 or 96.For a concrete illustration of the method, consider the case d = 16 and N = 4.One can show that the space M (Γ (4)) is five-dimensional, with the following basis.Let E ( z ) = 1 + 480 ∞ (cid:88) n =1 σ ( n ) q n be the Eisenstein series of weight 8 for SL ( Z ) (not to be confused with the E rootlattice), and let f be the newform of weight 8 for Γ (2) defined by f ( z ) = q ∞ (cid:89) n =1 (1 − q n ) (1 − q n ) . Then M g (Γ (4)) has the basis g , . . . , g , where g ( z ) = E ( z ), g ( z ) = 16 E (2 z ), g ( z ) = 256 E (4 z ), g ( z ) = f ( z ), and g ( z ) = 16 f (2 z ). The Atkin-Lehner involutionacts by (cid:101) g = g , (cid:101) g = g , (cid:101) g = g , (cid:101) g = g , and (cid:101) g = g . Using this information, wecan write down the linear program explicitly and solve it. As usual, the trickiestpart is identifying the right choice of T , while we can simply take M large enough(e.g., M = 10 is more than sufficient).For T = 2, solving the linear program yields the modular form (cid:88) j x j g j = 117 g + 117 g − g = 1 + 4320 q + 61440 q + 522720 q + · · · , which is the theta series of the Barnes-Wall lattice. Similarly, for T = 4 we obtain (cid:88) j x j g j = 1272 g + 1272 g − g = 1 + 4320 q + 61440 q + 522720 q + · · · , which is the same modular form with q replaced by q and which yields the samebound. For these two values of T , the space M (Γ (4)) is incapable of separatingthe linear programming bound from the center density 0 . T = 3 we obtain (cid:88) j x j g j = 1136 g − g + 1136 g − g − g = 1 + 7680 q + 4320 q + 276480 q + · · · , which yields an improved center density lower bound of 3 / = 0 . . . . ,more than 60% greater than the center density of the Barnes-Wall lattice. In fact,this modular form has been studied before: it is the extremal theta series in 16dimensions (see equation (47) in [13, p. 190]).It is tempting to conjecture that the extremal theta series should exactly matchthe optimal linear programming bound. This conjecture would be a beautifulanalogue of the behavior in 8 and 24 dimensions. In those cases the optimal latticeshave determinant 1 and minimal norm 2 or 4, respectively. The extremal thetaseries in 16 dimensions behaves like the theta series of a lattice of determinant 1 andminimal norm 3, exactly interpolating between 8 and 24 dimensions. Presumablyno such lattice exists, but the linear programming bound could match the densityof a hypothetical lattice.That is a good approximation in this case, but the answer turns out to be moresubtle: in Section 6, we obtain a better lower bound using N = 96. Instead ofminimal norm 3, the improved lower bound is 3 . . . . . , but we are unable to conjecture an exact formula for this number. UAL LINEAR PROGRAMMING BOUNDS FOR SPHERE PACKING. . . 11 Poisson summation analogues from modular forms
The main result of this section is Proposition 2.2, which yields a summationformula from a modular form. Summation formulas of this sort are well knownto number theorists, and essentially equivalent to the functional equation for the L -function. We record the details here and sketch a proof for the convenience of thereader. (One can also prove such a formula using the density of complex Gaussiansamong radial Schwartz functions, along the lines of Section 6 in [25] or Section 2.3in [11].)Proposition 2.2 is essentially a version of Voronoi summation. Our proof willfollow the approach used in standard proofs of Voronoi summation (for example, asin Section 10.2.5 of [4] or Section 2 of [22]). The key idea comes from the classicalobservation that the usual Poisson summation formula is a consequence of thefunctional equation of the Riemann zeta function. Similarly, Proposition 2.2 followsfrom the functional equation relating the L -functions associated to a modular formand its Atkin-Lehner dual.In what follows, we use the notation established in Proposition 2.2. To state thefunctional equation, we first define the L -function L ( s, g ) = ∞ (cid:88) n =1 a n n s when Re( s ) > k , and the completed L -function Λ( s, g ) = N s/ (2 π ) − s Γ( s ) L ( s, g ) . The functional equation relating Λ( s, g ) and Λ( s, (cid:101) g ) is classical, dating back to Hecke[19]. It says that the L -functions can be analytically continued so thatΛ( s, g ) + a s + b k − s is entire and bounded in every vertical strip, and we have the functional equationΛ( s, g ) = Λ( k − s, (cid:101) g ) , or equivalently(4.1) L (cid:16) k − s , g (cid:17) = N ( s − k ) / (2 π ) k − s Γ( s/ k − s/ L (cid:16) s , ˜ g (cid:17) . See, for example, Theorem 1 in [23, p. I-5].
Sketch of proof of Proposition 2.2.
For a radial Schwartz function f on R d , let S = (cid:88) n ≥ a n f ( √ n ) . By Mellin inversion, a n f ( √ n ) = 12 πi (cid:90) Re( s )= σ a n n s/ M f ( s ) ds for any σ >
0, where the
Mellin transform M f is defined by M f ( s ) = (cid:90) ∞ f ( x ) x s dxx . In particular, for σ = d + ε with ε > S = 12 πi ∞ (cid:88) n =1 (cid:90) Re( s )= d + ε a n n s/ M f ( s ) ds = 12 πi (cid:90) Re( s )= d + ε L (cid:16) s , g (cid:17) M f ( s ) ds, where switching the sum and integral is permitted because of the uniform convergenceof the sum defining the L -function.The integrand L ( s/ , g ) M f ( s ) is negligible when s has large imaginary part. Tosee why, note that by a stationary phase argument the Mellin transform M f ( s )is rapidly decaying as Im( s ) grows, while L ( s/ , g ) grows at most polynomiallyin Im( s ) by the Phragm´en-Lindel¨of principle. Thus, we can shift the contour ofintegration to the left, as long as we account for poles.It is not hard to check that M f ( s ) has a possible pole at s = 0 with residue f (0), L ( s/ , g ) has a possible pole at s = d with residue2 (cid:18) π √ N (cid:19) d/ d/ b , and L (0 , g ) = − a , since the pole of Γ( s ) at s = 0 cancels the pole of Λ( s, g ) at s = 0. Thus, S = − a f (0) + 2 b (cid:18) π √ N (cid:19) d/ d/ M f ( d ) + 12 πi (cid:90) Re( s )= − ε L (cid:16) s , g (cid:17) M f ( s ) ds. Setting T = 12 πi (cid:90) Re( s )= − ε L (cid:16) s , g (cid:17) M f ( s ) ds and applying the identity (cid:98) f (0) = π d/ Γ( d/ M f ( d ), we see that(4.2) a f (0) + S = (cid:18) √ N (cid:19) d/ b (cid:98) f (0) + T. Changing variables from s to d − s and applying the functional equation (4.1) yields T = 12 πi (cid:90) Re( s )= d + ε L (cid:18) d − s , g (cid:19) M f ( d − s ) ds = 12 πi (cid:90) Re( s )= d + ε N s/ − d/ (2 π ) d/ − s Γ( s/ d − s ) / L (cid:16) s , (cid:101) g (cid:17) M f ( d − s ) ds. Now we use the identity M (cid:98) f ( s ) = π d/ − s Γ( s/ d − s ) / M f ( d − s )(see Theorem 5.9 in [21]). Making this substitution, we find that T = (cid:18) √ N (cid:19) d/ πi (cid:90) Re( s )= d + ε (cid:18) N (cid:19) − s/ L (cid:16) s , (cid:101) g (cid:17) M (cid:98) f ( s ) ds. Replacing the L -function with its defining sum, switching the sum and integral asabove, and applying Mellin inversion again (reversing the steps from the start of UAL LINEAR PROGRAMMING BOUNDS FOR SPHERE PACKING. . . 13 the proof), we see that T = (cid:18) √ N (cid:19) d/ πi (cid:90) Re( s )= d + ε ∞ (cid:88) n =1 b n (4 n/N ) s/ M (cid:98) f ( s ) ds = (cid:18) √ N (cid:19) d/ ∞ (cid:88) n =1 πi (cid:90) Re( s )= d + ε b n (4 n/N ) s/ M (cid:98) f ( s ) ds = (cid:18) √ N (cid:19) d/ ∞ (cid:88) n =1 b n (cid:98) f (cid:18) √ n √ N (cid:19) . Hence, (4.2) implies that ∞ (cid:88) n =0 a n f ( √ n ) = (cid:18) √ N (cid:19) d/ ∞ (cid:88) n =0 b n (cid:98) f (cid:18) √ n √ N (cid:19) , as desired. (cid:3) Checking positivity of modular form coefficients
In this section, we explain how we check whether a modular form of weight k for Γ ( N ) has nonnegative coefficients in its q -series. This method uses onlystandard techniques from the theory of modular forms, but we describe them herefor the benefit of readers in discrete geometry. The key idea is that Eisensteinseries typically make the dominant contribution asymptotically, which reduces theproblem to a finite calculation if the Eisenstein contribution is positive.As mentioned above, we assume for simplicity that N is not divisible by 16 , 9 ,or p for any prime p >
3. This assumption guarantees that all the characters inthis section are real. Furthermore, we assume that k ≥
3, because the Eisensteinseries for weight 2 must be obtained using different formulas (the formulas thatwork for k ≥ k = 2).To verify that g = (cid:80) ∞ n =0 a n q n has a n ≥ n , we write g as g e + g c , where g e = (cid:80) ∞ n =0 e n q n is a linear combination of Eisenstein series and g c = (cid:80) ∞ n =0 c n q n iscuspidal, and we attempt to carry out the following steps:(1) Use Weil bounds to show that | c n | ≤ C g n k/ for some explicit constant C g .(2) Use explicit formulas for Eisenstein series to show that e n ≥ r g n k − forsome explicit constant r g > Q such that a n > n > Q .(4) Explicitly compute the coefficients a n of g to check that a n ≥ n ≤ Q .The first step is straightforward, given some powerful machinery. Deligne’s proofof the Weil conjectures [14] implies that, independent of weight, if h = (cid:80) ∞ n =1 c n q n is a cuspidal Hecke eigenform normalized so that c n (cid:48) = 1 for the minimal n (cid:48) with c n (cid:48) (cid:54) = 0, then | c n | ≤ σ ( n ) n ( k − / ≤ n k/ . Let B k ( N ) be the set of such eigenforms,which are a basis for the cuspidal part of M k (Γ ( N )). (Note that the elements of B k ( N ) typically do not have rational coefficients. Instead, we must work over alarger number field.) If g c = ∞ (cid:88) n =1 c n q n = (cid:88) h ∈ B k ( N ) x h h with coefficients x h ∈ C , then | c n | ≤ n k/ (cid:88) h ∈ B k ( N ) | x h | . Thus, step (1) holds with C g = (cid:80) h ∈ B k ( N ) | x h | .For the second step, we need to write down the Eisenstein series explicitly. Wecan describe them in terms of primitive Dirichlet characters φ of conductor u andnatural numbers t such that u t | N (where a | b means a divides b ). Thanks to ourdivisibility hypotheses on N , it follows that u |
24, and therefore φ must be a realcharacter; in other words, it takes on only the values ±
1. Then the Eisenstein seriesin M k (Γ ( N )) all have the form E φt = δ ( φ )2 L (1 − k, φ ) + (cid:88) n ≥ ,t | n φ ( n/t ) σ k − ( n/t ) q n , where σ (cid:96) ( m ) = (cid:80) d | m d (cid:96) , L ( s, φ ) is the L -function of φ , and δ ( φ ) = (cid:40) φ is the trivial character of conductor 1, and0 otherwise.See, for example, Theorem 4.5.2 in [16].Since the Eisenstein series span the Eisenstein part of M k (Γ ( N )), there existconstants y φt such that g e = (cid:88) t,φ y φt E φt = e + (cid:88) t,φ (cid:88) n ≥ ,t | n y φt φ ( n/t ) σ k − ( n/t )= e + ∞ (cid:88) n =1 (cid:88) t | N,t | n (cid:88) φ y φt φ ( n/t ) σ k − ( n/t ) . It is straightforward to check that whenever t | n , σ k − ( n ) σ k − ( t ) ≤ σ k − ( n/t ) ≤ σ k − ( n ) t k − . This implies that if we set r g ( t, n ) = (cid:40) t k − if (cid:80) φ y φt φ ( n/t ) <
0, and σ k − ( t ) if (cid:80) φ y φt φ ( n/t ) ≥ r g ( n ) = (cid:88) t | N,t | n (cid:88) φ y φt φ ( n/t ) r g ( t, n ) , and r g = min n ≥ r g ( n ) = min ≤ n ≤ N r g ( n ) , then e n ≥ σ k − ( n ) r g ≥ n k − r g . UAL LINEAR PROGRAMMING BOUNDS FOR SPHERE PACKING. . . 15
This completes step (2), provided that r g is positive. If it is not positive, then ourtest will be inconclusive, since we are unable to certify that even the Eisenstein partis nonnegative.Combining the results of the previous two steps, we find that a n ≥ n k − r g − n k/ C g . Since k >
2, this inequality provides an easily computed bound Q = (cid:98) ( C g /r g ) / ( k − (cid:99) such that a n > n > Q . Because of the large gap between n k − and n k/ ,the bound Q is typically relatively small. Finally, to certify that the coefficients of g are all nonnegative, we explicitly compute the coefficients a n for n ≤ Q .This method will not always work, without more careful estimates. For example,it fails if a n is not eventually positive. That can occur in practice: in the examplefrom Section 3 with d = 16, N = 4, and T = 2, the optimal modular form is g = 1 + 4320 q + 61440 q + 522720 q + 2211840 q + 8960640 q + · · · , which has eventually positive coefficients, but (cid:101) g = 16 + 69120 q + 983040 q + 8363520 q + 35389440 q + · · · , which does not. Thus, proving that (cid:101) g has nonnegative coefficients requires a littlemore care. However, we have not observed this phenomenon for the best choices of T in any of the cases we have examined. If it were to occur, it could be handled bydistinguishing between the values of r g ( n ) for different residue classes of n modulo N , and showing that the cuspidal contribution vanishes whenever r g ( n ) = 0.6. Numerical results
Table 6.1 shows our numerical results. We used the SageMath computer algebrasystem [26] for our calculations, with one exception: we used Magma [2] to computebases for modular forms and the action of the Atkin-Lehner involution. Thiscombination works conveniently, because SageMath has an interface for callingMagma code.To produce rigorous results, we used exact rational arithmetic, and we provednonnegativity of coefficients using the techniques of Section 5. For calculations withforms of level 24, we directly solved the linear program over Q ; for level 96, weinstead used floating point arithmetic to obtain an approximate solution, which wethen used to obtain a rational solution and prove its correctness and optimality. Allthe numbers in the table are rounded correctly: lower bounds are rounded down,and upper bounds are rounded up.7. Open problems
Our new lower bounds in Table 6.1 come fairly close to the known upper bounds,but they do not agree to many decimal places. We believe that the upper boundsagree with the true linear programming bound, aside from rounding the last decimalplace up, while the lower bounds could be further improved. One difficulty in doingso is that modular forms are inherently quantized: in the summation formula ∞ (cid:88) n =0 a n f ( √ n ) = (cid:18) √ N (cid:19) d/ ∞ (cid:88) n =0 b n (cid:98) f (cid:18) √ n √ N (cid:19) , Table 6.1.
Center density bounds in dimensions 8 through 32.The upper bound is the linear programming bound, computedusing the best auxiliary function currently known [7], while thedual bound is based on the given values of N and T , and the recordpacking is the densest packing currently known [13]. In dimensions12 and 16, we include both N = 96 and N = 24 for comparison.Dimension Record packing Dual bound Upper bound N T . . . . . . . . . . . . . . . . . . . √ n or 2 (cid:112) n/N slightly, and so one mustdo the best one can using only radii of these forms. In particular, closely matchingthe upper bound may require N to be very large, perhaps on the order of 10 ifwe wish to match ten digits, and dealing with such large N is not practical. Anyfeasible method that could close the gap between the primal and dual bounds towithin a factor of 1 + 10 − would be a significant advance, and modular forms mightnot be the right tool for this purpose. For comparison, [30] and [27] obtain duallinear programming bounds in high dimensions using an entirely different approach.Another topic we leave open is computations in dimensions that are not divisibleby 4. We see no theoretical obstacle to such an extension: one must simply usemodular forms of odd weight (for dimensions divisible by 2 but not 4) or half-integralweight (for odd dimensions), and replace Γ ( N ) with Γ ( N ) so that such forms exist.However, we have not implemented these computations. We have also not exploredthe uncertainty principle introduced in [3] and further studied in [7], for which onecould again prove dual bounds using modular forms.One intriguing possibility that may be nearly within reach is proving that thereexists a dimension in which the linear programming bound is not sharp. Alldimensions except 1, 2, 8, and 24 seem to have this property, but so far no proof isknown. Three dimensions would be a natural target, because we know the optimalpacking density, and thus it would suffice to prove any dual bound greater thanthis density. In higher dimensions, it would require an improvement on the linearprogramming bound. The only such bound currently known is Theorem 1.4 fromde Laat, Oliveira, and Vallentin’s paper [20], which is a refinement of the linearprogramming bound that seems to give a small numerical improvement in dimensions3, 4, 5, 6, 7, and 9 (see Table 1 in [20]) and presumably higher dimensions as well,aside from 24. Any dual bound greater than this improved upper bound wouldsuffice to show that the linear programming bound is not sharp. Conversely, itwould be interesting to prove dual bounds for the theorem of de Laat, Oliveira, andVallentin itself. UAL LINEAR PROGRAMMING BOUNDS FOR SPHERE PACKING. . . 17
References [1] G. E. Andrews, R. Askey, and R. Roy,
Special Functions , Encyclopedia of Mathematics andits Applications , Cambridge University Press, Cambridge, 1999.doi:10.1017/CBO9781107325937 MR1688958[2] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system I: the user language ,J. Symbolic Comput. (1997), no. 3–4, 235–265. doi:10.1006/jsco.1996.0125 MR1484478[3] J. Bourgain, L. Clozel, and J.-P. Kahane, Principe d’Heisenberg et fonctions positives , Ann.Inst. Fourier (Grenoble) (2010), no. 4, 1215–1232. doi:10.5802/aif.2552 MR2722239[4] H. Cohen, Number Theory. Volume II: Analytic and Modern Tools , Graduate Texts inMathematics , Springer, New York, 2007. doi:10.1007/978-0-387-49894-2 MR2312338[5] H. Cohn,
New upper bounds on sphere packings II , Geom. Topol. (2002), 329–353.arXiv:math/0110010 doi:10.2140/gt.2002.6.329 MR1914571[6] H. Cohn and N. Elkies, New upper bounds on sphere packings I , Ann. of Math. (2) (2003),no. 2, 689-714. arXiv:math/0110009 doi:10.4007/annals.2003.157.689 MR1973059[7] H. Cohn and F. Gon¸calves,
An optimal uncertainty principle in twelve dimensions via modularforms , Invent. Math. (2019), no. 3, 799–831. arXiv:1712.04438 doi:10.1007/s00222-019-00875-4 MR3989254[8] H. Cohn and A. Kumar,
Universally optimal distribution of points on spheres , J. Amer.Math. Soc. (2007), no. 1, 99–148. arXiv:math/0607446 doi:10.1090/S0894-0347-06-00546-7MR2257398[9] H. Cohn and A. Kumar, Optimality and uniqueness of the Leech lattice among lattices , Ann. ofMath. (2) (2009), no. 3, 1003–1050. arXiv:math/0403263 doi:10.4007/annals.2009.170.1003MR2600869[10] H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska,
The sphere packingproblem in dimension
24, Ann. of Math. (2) (2017), no. 3, 1017–1033. arXiv:1603.06518doi:10.4007/annals.2017.185.3.8 MR3664817[11] H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska,
Universal optimality ofthe E and Leech lattices and interpolation formulas , preprint, 2019. arXiv:1902.05438[12] H. Cohn and Y. Zhao, Sphere packing bounds via spherical codes , Duke Math. J. (2014),no. 10, 1965–2002. arXiv:1212.5966 doi:10.1215/00127094-2738857 MR3229046[13] J. H. Conway and N. J. A. Sloane,
Sphere Packings, Lattices and Groups , third edition,Grundlehren der Mathematischen Wissenschaften , Springer-Verlag, New York, 1999.doi:10.1007/978-1-4757-6568-7 MR1662447[14] P. Deligne,
La conjecture de Weil. I , Inst. Hautes ´Etudes Sci. Publ. Math. (1974), 273–307. MR0340258[15] F. Diamond and J. Im,
Modular forms and modular curves , in V. K. Murty, ed.,
Seminar onFermat’s Last Theorem (Toronto, ON, 19931994) , pp. 39–133, CMS Conf. Proc. , Amer.Math. Soc., Providence, RI, 1995. MR1357209[16] F. Diamond and J. Shurman, A First Course in Modular Forms , Graduate Texts in Mathe-matics , Springer-Verlag, New York, 2005. doi:10.1007/978-0-387-27226-9 MR2112196[17] T. C. Hales,
A proof of the Kepler conjecture , Ann. of Math. (2) (2005), no. 3, 1065–1185.doi:10.4007/annals.2005.162.1065 MR2179728[18] T. Hales, M. Adams, G. Bauer, T. D. Dang, J. Harrison, L. T. Hoang, C. Kaliszyk, V. Magron,S. McLaughlin, T. T. Nguyen, Q. T. Nguyen, T. Nipkow, S. Obua, J. Pleso, J. Rute,A. Solovyev, T. H. A. Ta, N. T. Tran, T. D. Trieu, J. Urban, K. Vu, and R. Zumkeller,
A formal proof of the Kepler conjecture , Forum Math. Pi (2017), e2, 29 pp. arXiv:1501.02155doi:10.1017/fmp.2017.1 MR3659768[19] E. Hecke, ¨Uber die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung , Math.Ann. (1936), no. 1, 664–699. doi:10.1007/BF01565437 MR1513069[20] D. de Laat, F. M. de Oliveira Filho, and F. Vallentin, Upper bounds for packings ofspheres of several radii , Forum Math. Sigma (2014), e23, 42 pp. arXiv:1206.2608doi:10.1017/fms.2014.24 MR3264261[21] E. H. Lieb and M. Loss, Analysis , second edition, Graduate Studies in Mathematics ,American Mathematical Society, Providence, RI, 2001. doi:10.1090/gsm/014 MR1817225[22] S. D. Miller and W. Schmid, Summation formulas, from Poisson and Voronoi to the present ,in
Noncommutative Harmonic Analysis , Progr. Math. , 419–440, Birkh¨auser Boston,Boston, MA, 2004. arXiv:math/0304187 doi:10.1007/978-0-8176-8204-0 15 MR2036579 [23] A. Ogg,
Modular Forms and Dirichlet Series , W. A. Benjamin, Inc., New York-Amsterdam,1969. MR0256993[24] M. Ohta,
On the p -adic Eichler-Shimura isomorphism for Λ -adic cusp forms , J. Reine Angew.Math. (1995), 49–98. doi:10.1515/crll.1995.463.49 MR1332907[25] D. Radchenko and M. Viazovska, Fourier interpolation on the real line , Publ. Math. Inst.Hautes ´Etudes Sci. (2019), 51–81. arXiv:1701.00265 doi:10.1007/s10240-018-0101-zMR3949027[26]
SageMath, the Sage Mathematics Software System (Version 8.8) , The Sage Developers, 2019, .[27] A. Scardicchio, F. H. Stillinger, and S. Torquato,
Estimates of the optimal density of spherepackings in high dimensions , J. Math. Phys. (2008), no. 4, 043301, 15 pp. arXiv:0705.1482doi:10.1063/1.2897027 MR2412293[28] W. Stein, Modular Forms, a Computational Approach , Graduate Studies in Mathematics ,American Mathematical Society, Providence, RI, 2007 doi:10.1090/gsm/079 MR2289048[29] A. Thue, Om nogle geometrisk-taltheoretiske Theoremer , Forhandlingerne ved de SkandinaviskeNaturforskeres (1892), 352–353.[30] S. Torquato and F. H. Stillinger, New conjectural lower bounds on the optimal density ofsphere packings , Experiment. Math. (2006), no. 3, 307–331. arXiv:math/0508381 https://projecteuclid.org/euclid.em/1175789761 MR2264469[31] M. S. Viazovska,
The sphere packing problem in dimension
8, Ann. of Math. (2) (2017),no. 3, 991–1015. arXiv:1603.04246 doi:10.4007/annals.2017.185.3.7 MR3664816
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