New upper bounds for spherical codes and packings
aa r X i v : . [ m a t h . M G ] O c t NEW UPPER BOUNDS FOR SPHERICAL CODES AND PACKINGS
NASER TALEBIZADEH SARDARI AND MASOUD ZARGAR
Abstract.
We improve the previously best known upper bounds on the sizes of θ -spherical codesfor every θ < θ ∗ ≈ . ◦ at least by a factor of 0 . n ≥ . n . Our method also breaks many non-numerical sphere packing density boundsin small dimensions. Apart from Cohn and Zhao’s [CZ14] improvement on the geometric averageof Levenshtein’s bound [Lev79a] over all sufficiently high dimensions by a factor of 0 . , our workis the first improvement for each dimension since the work of Kabatyanskii and Levenshtein [KL78]and its later improvement by Levenshtein [Lev79a]. Moreover, we generalize Levenshtein’s opti-mal polynomials and provide explicit formulae for them that may be of independent interest. For0 < θ < θ ∗ , we construct a test function for Delsarte’s linear programing problem for θ -sphericalcodes with exponentially improved factor in dimension compared to previous test functions.
1. Introduction 12. Orthogonal polynomials 113. Geometric improvement 124. New test functions 145. Comparison with previous bounds 186. Local approximation of Jacobi polynomials 267. Conditional density functions 278. The critical function 319. Critical value of functional 3510. Positivity of Fourier coefficients 3611. Numerics 38References 401.
Introduction
Sphere packings.
Packing densities have been studied extensively, for purely mathematicalreasons as well as for their connections to coding theory. The work of Conway and Sloane is acomprehensive reference for this subject [CS99]. We proceed by defining the basics of this subject.Consider R n equipped with the Euclidean metric | . | and the associated volume vol( . ). For each real r > x ∈ R n , we denote by B n ( x, r ) the open ball in R n centered at x and of radius r .For each discrete set of points S ⊂ R n such that any two distinct points x, y ∈ S satisfy | x − y | ≥ P := ∪ x ∈ S B n ( x, , the union of non-overlapping unit open balls centered at the points of S . This is called a spherepacking ( S may vary), and we may associate to it the function mapping each real r > δ P ( r ) := vol( P ∩ B n (0 , r ))vol( B n (0 , r )) . he packing density of P is defined as δ P := lim sup r →∞ δ P ( r ) . Clearly, this is a finite number. The maximal sphere packing density in R n is defined as δ n := sup P ⊂ R n δ P , a supremum over all sphere packings P of R n by non-overlapping unit balls.The linear programming method initiated by Delsarte is a powerful tool for giving upper boundson sphere packing densities [Del72]. That being said, we only know the optimal sphere packingdensities in dimensions 1,2,3,8 and 24 [FT43, Hal05, Via17, CKM + δ = 1. In dimension 2, the best sphere packing is achieved by the usual hexagonallattice packing with δ = π/ √
12. A rigorous proof was provided by L. Fejes T´oth in 1943 [FT43];however, a proof was also given earlier by A.Thue in 1910 [Thu10], but it was considered incompleteby some experts in the field. In dimension 3, this is the subject of the Kepler conjecture, and wasresolved in 1998 by T.Hales [Hal05]. As a result of his work, we know that δ = π/ √
18. The othertwo known cases of optimal sphere packings were famously resolved in dimensions 8 and 24 by M.Viazovska and her collaborators in 2016. Based on some of the ideas of Cohn and Elkies [CE03],M. Viazovska first resolved the dimension 8 case [Via17]. Shortly afterward, she along with Cohn,Kumar, Miller, and Radchenko resolved the case of 24 dimensions [CKM + E -lattice with δ = π / δ = π / (12!). All theseoptimal sphere packings come from even unimodular lattices. Very recently, the first author provedan optimal upper bound on the sphere packing density of all but a tiny fraction of even unimodularlattices in high dimensions; see [Sar19, Theorem 1.1].The best known upper bounds on sphere packing densities in low dimensions are based on thelinear programming method developed by Cohn and Elkies [CE03] which itself is inspired by Del-sarte’s linear programing method. More recently the upper bounds have been improved in lowdimensions using the semi-definite linear programing method [dLdOFV14]. However, in high di-mensions, the most successful method to date for obtaining asymptotic upper bounds goes back toKabatyanskii–Levenshtein from 1978 [KL78]. This method is based on first bounding from abovethe maximal size of spherical codes using Delsarte’s linear programming method. We discuss thisin the next subsection.1.2. Spherical codes.
A notion closely related to sphere packings of Euclidean spaces is that ofspherical codes. Given S n − , the unit sphere in R n , a θ - spherical code is a finite subset A ⊂ S n − such that no two distinct x, y ∈ A are at an angular distance less than θ . For each 0 < θ ≤ π , wedefine M ( n, θ ) to be the largest cardinality of a θ -spherical code A ⊂ S n − . Suppose that p α,βm ( t ) is, up to normalization, the Jacobi polynomial of degree m with parame-ters ( α, β ). We denote by t α,β ,m its largest root. Levenshtein proved the following inequality usingDelsarte’s linear programing method. See Section 2 for a summary of results on Jacobi polynomials. Theorem 1.1 (Levenshtein, Theorem 6.2 of [Lev98]) . If cos θ ≤ t α +1 ,α + ε ,ℓ − ε for some < θ < π/ ,some ℓ , ε ∈ { , } , and α := n − , then M ( n, θ ) ≤ (cid:18) ℓ + n − n − (cid:19) + (cid:18) ℓ + n − − εn − (cid:19) . his is a less refined version of an upper bound proved by Levenshtein where the right hand sideis given by the value of a functional applied to a polynomial depending on θ . We will shortlydiscuss this functional in the general setting of this paper. This theorem of Levenshtein waspreceeded by a weaker theorem of Kabatyanskii–Levenshtein from 1978 [KL78].The bounds onsphere packing densities follow from such bounds on spherical codes via an elementary geometricargument that allows to relate sphere packings in R n to spherical codes. Indeed, for any 0 < θ ≤ π/ δ n ≤ M ( n + 1 , θ ) µ n ( θ ) , where µ n ( θ ) := 1 / sin n ( θ/ θ ∗ := 62 . ... ◦ be the unique root of the equation cos θ log( θ − sin θ ) − (1 + cos θ ) sin θ = 0. In [KL78], Kabatyanskii and Levenshtein proved the following bound by usinginequality (1) for θ ∗ and a weaker form of Theorem 1.1 [KL78, (52)] to bound M ( n, θ ∗ ) (it givesthe same exponent 0.599 as using Theorem 1.1). Theorem 1.2 (Kabatyanskii–Levenshtein, 1978) . As n → ∞ , δ n ≤ − . n (1+ o (1)) . Let 0 ≤ θ < θ ′ ≤ π . We write µ n ( θ, θ ′ ) for the mass of the cap with radius sin( θ/ θ ′ / on the unitsphere S n − (with respect to the natural probability measure). The best known bounds on M ( n, θ )for θ < θ ∗ are obtained via a similar elementary argument of Sidelnikov [Sid74] stating that for0 < θ < θ ′ ≤ π (2) M ( n, θ ) ≤ M ( n + 1 , θ ′ ) µ n ( θ, θ ′ ) . Indeed Barg and Musin [BM07, p.11 (8)], based on the work [AVZ00] of Agrell, Vargy, and Zeger,improved the above inequality and showed that(3) M ( n, θ ) ≤ M ( n − , θ ′ ) µ n ( θ, θ ′ ) . whenever π > θ ′ > (cid:16)
12 cos( θ/ (cid:17) . As demonstrated by Kabatyanskii–Levenshtein in [KL78],for large dimensions n and 0 < θ < θ ∗ the linear programming upper bound on M ( n, θ ) is exponen-tially weaker than the one obtained from the combination of the linear programming upper boundon M ( n, θ ∗ ) with inequality (2) for M ( n, θ ). Cohn and Zhao [CZ14] improved sphere packing den-sity upper bounds by combining the above upper bound of Kabatyanskii–Levenshtein on M ( n, θ )with their analogue of (3) when θ → π/ ≤ θ ≤ π ,(4) δ n ≤ M ( n, θ ) µ n ( θ ) , leading to better bounds than those obtained from (1). Thus, their improvement is a consequenceof a geometric argument at the level of comparing sphere packings to spherical codes. Our firstresult removes the angular restrictions on the Barg and Musin result. In the following, let s := cos θ , s ′ := cos θ ′ , r := q s − s ′ − s ′ , γ θ,θ ′ := 2 arctan s p (1 − s )( s − s ′ ) + arccos( r ) − π, and R := cos( γ θ,θ ′ ) . roposition 1.3. Let < θ < θ ′ < π. We have (5) M ( n, θ ) ≤ M ( n − , θ ′ ) µ n ( θ, θ ′ ) (1 + O ( ne − nc )) , where c := log (cid:16) − r − R (cid:17) > is independent of n and only depends on θ and θ ′ . We prove this Proposition in Section 3. A completely analogous result removes the restriction θ ≥ π/ Remark . Asymptotically as n → ∞ , Proposition 1.3 combined with the asymptotic bound onspherical codes due to Kabatyanskii–Levenshtein improves the best bound on M ( n, θ ) on (geo-metric) average by a factor 0 . ∼ (0 . for every θ < θ ′ and θ ′ > (cid:16)
12 cos( θ/ (cid:17) . Thisimprovement for spherical codes is the square of the improvement of Cohn and Zhao for the spherepacking density.Our main theorem regarding spherical codes is a linear programming variant of the inequality (5)which we state in the next section. The variant of inequality (5) is improved with an extra factor0 . n (rather than averaging over all high dimensions). In the caseof sphere packings, we obtain the same asymptotic improvement. Furthermore, we show that fordimensions n ≥ . n . As a consequence, weobtain a constant improvement to all previously known linear programming bounds on sphericalcodes and sphere packing densities. Our improvement does not lead to an improvement of theexponent 0 .
599 in Theorem 1.2; this exponent is to this day the best known. That being said,our geometric ideas combined with numerics lead to improvements that are better than the 0 . Constant improvement.
First, we discuss Delsarte’s linear programming method, and thenwe state our main theorem.Throughout this paper, we work with probability measures µ on [ − , . µ gives an inner prod-uct on the space of real polynomials R [ t ], and let { p i } ∞ i =0 be an orthonormal basis with respect to µ such that the degree of p i is i and p i (1) > i . Note that p = 1. Such a basis is uniquelydetermined by µ . Suppose that the basis elements p k define positive definite functions on S n − ,namely(7) X x i ,x j ∈ A h i h j p k ( h x i , x j i ) ≥ A ⊂ S n − and any real numbers h i ∈ R . An example of a probability measuresatisfying inequality (7) is dµ α := (1 − t ) α R − (1 − t ) α dt dt, where α ≥ n − and 2 α ∈ Z . Given s ∈ [ − , D ( µ, s ) of all functions f ( t ) = P ∞ i =0 f i p i ( t ), f i ∈ R , such that(1) f i ≥ f > , (2) f ( t ) ≤ − ≤ t ≤ s. Suppose 0 < θ < π , and A = { x , . . . , x N } is a θ -spherical code in S n − . Given a function f ∈ D ( µ, cos θ ), we consider X i,j f ( h x i , x j i ) . his may be written in two different ways as N f (1) + X i = j f ( h x i , x j i ) = f N + ∞ X k =1 f k X i,j p k ( h x i , x j i ) . Since f ∈ D ( µ, cos θ ) and h x i , x j i ≤ cos θ for every i = j , this gives us the inequality N ≤ f (1) f . We define L ( f ) := f (1) f . In particular, this method leads to the linear programming bound(8) M ( n, θ ) ≤ inf f ∈ D ( dµ n − , cos θ ) L ( f ) . Levenshtein proved Theorem 1.1 by introducing a family of even and odd degree polynomials inside D ( dµ n − , cos θ ) that minimize L ( f ) among all polynomials of smaller degrees inside D ( dµ n − , cos θ ).We call them Levenshtein’s polynomials. Let g θ ′ ∈ D ( dµ n − , cos θ ′ ) be the Levenshtein polynomialassociated to angle θ ′ and dimension n −
1. As we pointed out, for θ < θ ∗ the best known upperbounds so far are obtained from a combination of variants of inequality (2) or our (5) to linearprogramming bounds (8) for the angle θ ∗ , at least in high dimensions. However, it is possible todirectly provide a function inside D ( dµ n − , cos θ ) for θ < θ ∗ that improves this bound withoutrelying on inequality (5). We now state one of our main theorems. Theorem 1.4.
Fix θ < θ ∗ . Suppose < θ < θ ′ ≤ π/ , where cos( θ ′ ) = t α +1 ,α + ε ,d − ε for some ε ∈ { , } and α := n − . Then there is a function h ∈ D ( dµ n − , cos θ ) such that L ( h ) ≤ c n L ( g θ ′ ) µ n ( θ, θ ′ ) , where c n ≤ . for large enough n independent of θ and θ ′ . Corollary 1.5.
Fix < θ < θ ∗ . There exists a solution to Delsarte’s linear programing problemfor θ -spherical codes with exponentially improved factor in dimension compared to Levenshtein’soptimal polynomials for θ -spherical codes.Proof. By [KL78, Theorem 4]lim n →∞ n log L ( g θ ) = 1 + sin θ θ log 1 + sin θ θ − − sin θ θ log 1 − sin θ θ . Let h ∈ D ( dµ n − , cos θ ), which minimizes L ( h ) . By the first part of Theorem 1.4, we havelim n →∞ n log L ( h ) ≤ inf π/ ≥ θ ′ >θ (cid:18) θ ′ θ ′ log 1 + sin θ ′ θ ′ − − sin θ ′ θ ′ log 1 − sin θ ′ θ ′ (cid:19) − lim n →∞ log µ n ( θ, θ ′ ) n . It is easy to show that [KL78, Proof of Theorem 4]lim n →∞ log µ n ( θ, θ ′ ) n = 12 log 1 − cos θ − cos θ ′ . Hence, lim n →∞ n log L ( h ) ≤ inf π/ ≥ θ ′ >θ ∆( θ ′ ) −
12 log(1 − cos θ ) . where ∆( θ ′ ) := 1 + sin θ ′ θ ′ log 1 + sin θ ′ θ ′ − − sin θ ′ θ ′ log 1 − sin θ ′ θ ′ + 12 log(1 − cos θ ′ ) . ote that ddθ ′ ∆( θ ′ ) = cos θ ′ log( 1 + sin θ ′ − sin θ ′ ) − (1 + cos θ ′ ) sin θ ′ . As we mentioned above, θ ∗ := 62 . ... ◦ is the unique root of the equation ddθ ′ ∆( θ ′ ) = 0 [KL78,Theorem 4] in the interval 0 < θ ′ ≤ π/
2, which is a unique minimum of ∆( θ ′ ) for 0 < θ ′ ≤ π/ θ < θ ∗ lim n →∞ n (log L ( h ) − log L ( g θ )) ≤ ∆( θ ∗ ) − ∆( θ ) < . This concludes the proof of our corollary. (cid:3)
Finding functions h ∈ D ( dµ n − , cos θ ) with smaller value L ( h ) than L ( g θ ) for Levenshtein’s poly-nomials had been suggested by Levenshtein in [Lev98, page 117]. In fact, Boyvalenkov–Danev–Bumova [BDB96] gives necessary and sufficient conditions for constructing extremal polynomials that improve Levenshtein’s bound. However, their construction does not improve the exponentof Levenshtein’s bound as the asymptotic degrees of their polynomials are the same as those ofLevenshtein’s polynomials.We now give a uniform version of our theorem for the Sphere packing problem. Theorem 1.6.
Suppose that / ≤ cos( θ ′ ) < / . We have δ n − ≤ c n − ( θ ′ ) L ( g θ ′ ) µ n − ( θ ′ ) , where c n ( θ ′ ) < for every n . If, additionally, n ≥ and cos( θ ′ ) = t α +1 ,α + ε ,d − ε for some ε ∈ { , } ,where α = n − , we have c n ( θ ′ ) ≤ .
515 + n . Furthermore, for n ≥ we have c n ( θ ∗ ) ≤ . n . Note that by Kabatyanskii–Levenshtein [KL78], the best angle for such a comparison is asymptot-ically θ ∗ as comparisons using other angles are exponentially worse. Consequently, this theoremimplies that we have an improvement by 0 . c n ( θ ′ ) are non-trivially boundedfrom above uniformly in θ ′ . Remark . We prove Theorem 1.6 by constructing a test function that satisfies the Cohn–Elkieslinear programing conditions. This construction is based on the work of Cohn and Zhao [CZ14]which proves the above theorem for c n = 1. The factor 0 . n is the constant improvementover the combination of the work of Cohn and Zhao [CZ14] with Levenshtein’s optimal polynomi-als [Lev79a]. Furthermore, note that the right hand side corresponds to the dimension n −
1, andso the dimension of the right hand side does not increase as happens in the case of Sidelnikov’sinequality (2).We begin the proofs of Theorems 1.4 and 1.6 in Section 4 and complete them in Section 5. We startthe construction of h in Section 4. In Proposition 4.6, we prove a general version of Theorem 1.4. Weconstruct h for every pair of angles 0 < θ < θ ∗ ≤ θ ′ < π/ g θ ′ ∈ D ( dµ n − , cos θ ′ ) for c n = 1 . In particular, Proposition 1.3 and its linear programming version Proposition 4.6 generalizethe construction of Cohn and Zhao which works only for θ ≈ θ ′ ≥ π/ h with c n < h with c n < . Our construction isbased on estimating the triple density functions of points in Section 7 and estimating the Jacobipolynomials near their extreme roots in Section 6. It is known that the latter problem is very diffi-cult [Kra07, Conjecture 1]. In Section 6, we overcome this difficulty by using the relation between Rogers Levenshtein79 K.–L. Cohn–Zhao C.–Z.+L79 New bound12 8 . × − . × − . × . × − . × − . × −
24 2 . × − . × − . × − . × − . × − . × −
36 5 . × − . × − . × − . × − . × − . × −
48 1 . × − . × − . × − . × − . × − . × −
60 2 . × − . × − . × − . × − . × − . × −
72 4 . × − . × − . × − . × − . × − . × −
84 7 . × − . × − . × − . × − . × − . × −
96 1 . × − . × − . × − . × − . × − . × −
108 2 . × − . × − . × − . × − . × − . × −
120 3 . × − . × − . × − . × − . × − . × − Table 1.
Upper bounds on maximal sphere packing densities δ n in R n . The lastcolumn is obtained from Proposition 4.7 by minimizing the right hand side of thisproposition as we vary the angle θ between π/ π and maximize δ > h toobtain the last column of Table 1, which already gives better improvement in low dimensions thanour asymptotic result. We now describe the different columns. The Rogers column corresponds tothe bounds on sphere packing densities obtained by Rogers [Rog58]. The
Levenshtein79 column cor-responds to the bound obtained by Levenshtein in terms of roots of Bessel functions [Lev79a]. The
K.–L. column corresponds to the bound on M ( n, θ ) proved by Kabatyanskii and Levenshtein [KL78]combined with Sidelnikov’s inequality (2). The Cohn–Zhao column corresponds to the columnfound in the work of Cohn and Zhao [CZ14]; they combined their inequality (4) with the boundon M ( n, θ ) proved by Kabatyanskii–Levenshtein. We also include the column C.–Z.+L79 whichcorresponds to combining Cohn and Zhao’s inequality with improved bounds on M ( n, θ ) using Lev-enshtein’s optimal polynomials [Lev79a]. The final column corresponds to the bounds on spherepacking densities obtained by our method. Note that our bounds break most of the other boundsalso in smaller dimensions. Another advantage is that we provide an explicit function satisfyingthe Cohn–Elkies linear programming conditions. Our bounds come from this explicit function andonly involve explicit integral calculations; in contrast to the numerical method in [CE03], we donot rely on any searching algorithm. Moreover, compared to the Cohn–Elkies linear programmingmethod, in n = 120 dimensions, we substantially break the sphere packing density upper bound of1 . × − obtained by forcing eight double roots.1.4. Generalizations of Levenshtein’s extremal polynomials.
In addition to proving the re-sults mentioned in the previous subsections for θ < θ ∗ , we end this paper by providing a newconceptual derivation of Levenshtein’s extremal polynomials of both even and odd degrees. This isorthogonal to what was discussed in the previous subsections. Moreover, we derive explicit closedformulae for generalized version of Levenshtein’s extremal polynomials in addition to explicit for-mulae for the value of the functional L on such extremal functions.More precisely, we study a problem more general than what has been studied in the literatureon optimizing Levenshtein’s polynomials. Indeed, we introduce the spaces Λ µ,d,η , associated tosome continuous function η , whose elements satisfy the second condition of the definition of D ( µ, s )trivially. We then find the infimum of the functional L over Λ µ,d,η and show that this infimum is chieved by a function that lies in Λ µ,d,η ∩ D ( µ, s ) under some explicit conditions. We recover Lev-enshtein’s extremal polynomials of odd and even degrees which correspond to η = 1 and η = 1 + t ,respectively.We now precisely define the spaces Λ µ,d,η , and state our first theorem regarding the critical func-tions of the functional L . Suppose that η ( t ) is a continuous non-negative function on [ − , , where R − η ( t ) dt > . For example we may take η ( t ) to be a polynomial which is positive on [ − , η ( t ) = 1 + t. Let Λ µ,d,η be the space of all functions g ( t ) = ( t − s ) η ( t ) f ( t ) , where f ( t ) is somepolynomial of degree at most d and g >
0. Let { ˜ p i } be the orthonormal basis of polynomials withrespect to the measure η ( t ) µ where ˜ p i has degree i and ˜ p i (1) >
0. For example for η ( t ) = 1, wehave ˜ p i = p i . Let f [ µ,d,η ] ( t ) := d − X i =0 λ c i t − s det (cid:20) ˜ p i +1 ( t ) ˜ p i +1 ( s )˜ p i ( t ) ˜ p i ( s ) (cid:21) , where λ c i := a i +1 (cid:16) ˜ p i (1)˜ p i ( s ) − ˜ p i +1 (1)˜ p i +1 ( s ) (cid:17) (˜ a i depends on the basis { ˜ p n } , and is defined in equation (17)in Section 2). Let g [ µ,d,η ] ( t ) := ( t − s ) η ( t ) f [ µ,d,η ] ( t ) , and denote by d ( s ) and d ( s ) the first andsecond positive integers i such that ˜ p i ( s )˜ p i +1 ( s ) <
0. Note that we are suppressing s from some ofour notations. Next, we state our main theorem. Theorem 1.7. g [ µ,d,η ] ( t ) (up to a positive scalar multiple) is the unique critical point of L over g ∈ Λ µ,d,η provided that g [ µ,d,η ]0 > . Moreover, g [ µ,d,η ] ( t ) (up to a positive scalar multiple) is theunique minimum of L over g ∈ Λ µ,d,η provided that d ( s ) ≤ d < d ( s ) and g [ µ,d,η ]0 > . We prove Theroem 1.7 in Section 8. The argument is based on Lagrange multiplier method anddiagonalizing a quadratic from which is the zeroth Fourier coefficients (in the basis p i ) and so wecannot continue to work with the basis p i because of the presence of η ( t ). This is why we workin the basis ˜ p i , an orthonormal basis of R [ t ] with respect to the measure η ( t ) dµ ( t ). Once we havesuch a diagonalization, the condition d ( s ) ≤ d < d ( s ) ensures a signature property which inturn ensures global minimality of the critical function g [ µ,d,η ] assuming the positivity of the zerothFourier coefficient in the expansion in the basis { p i } . Note that Λ µ,d,η is not a subspace of D ( µ, s );we need to choose µ, d, η (depending on s ) appropriately so that g [ µ,d,η ] is in D ( µ, s ) . We discussthis in the following subsection.1.5.
Positivity of Fourier coefficients.
Once we obtain the critical function g [ µ,d,η ] as aboveand obtain the strict positivity of the zeroth Fourier coefficient, we need to give conditions underwhich the other Fourier coefficients are non-negative. As a result, we restrict µ and η ( t ) to havethe following positivity properties. Definition 1.8.
We say µ satisfies the Krein condition if for every i, j, k ≥ , Z − p i ( t ) p j ( t ) p k ( t ) dµ ≥ . Note that dµ α satisfies the Krein condition; see [KL78, Equation (38)]. Definition 1.9.
We say a continuous function η is ( µ, s, d ) -positive if there exists a non-zero c ∈ C such that for every i, j ≥ , (10) c Z − p i ( t )( t − s ) η ( t ) f [ µ,d,η ] ( t ) dµ ≥ , nd(11) c Z − p j ( t ) f [ µ,d,η ] ( t ) dµ ≥ . We often use the following expressions for f [ µ,d,η ] ( t ) in order to check ( µ, s, d )-positivity condition.One may view these formulas as the generalized Christoffel-Darboux formula.
Theorem 1.10.
We have f [ µ,d,η ] ( t ) = ˜ c ( t − t − s ) det ˜ p d +1 ( t ) ˜ p d +1 ( s ) ˜ p d +1 (1)˜ p d ( t ) ˜ p d ( s ) ˜ p d (1)˜ p d − ( t ) ˜ p d − ( s ) ˜ p d − (1) for some ˜ c ∈ C . Moreover, f [ µ,d,η ] ( t ) = ˜ c ′ d − X i =0 ˜ p i ( t ) det (cid:20) ˜ p d ( s ) ˜ p d (1)˜ p i ( s ) ˜ p i (1) (cid:21) for some ˜ c ′ ∈ C . Note that condition (1) for g [ µ,d,η ] ( t ) = ( t − s ) η ( t ) f [ µ,d,η ] ( t ) , is an infinite system of quadraticinequalities in terms of the coefficients of f [ µ,d,η ] ( t ) , and the quadratic forms depend on the multi-plicative structure of the Jacobi polynomials. So, checking condition (1) directly for g [ µ,d,η ] is veryhard. We have the following criteria, though sufficient, is not a necessary condition for (1). Theorem 1.11.
Suppose that µ satisfies the Krein condition and η is ( µ, s, d ) -positive. Then g [ µ,d,η ] ∈ D ( µ, s ) . Polynomial η and comparison with other works. Next, we consider special examples of g [ µ,d,η ] ( t ), where η is a polynomial. In particular, we consider µ = dµ α := (1 − t ) α dt/ R − (1 − t ) α dt with α > − η = 1 or η = 1 + t. These examples are closely related to Levenshtein’s extremalpolynomials with odd and even degrees respectively; see [Lev98, Lev92, Lev79b]. The extremalproperties of these polynomials has been studied extensively in the works of Boyvalenkov [Boy95,BDH +
19] and also the work of Barg and Nogin [BN08]. Suppose that η ( t ) ∈ R [ t ] is a polynomialwith roots α , . . . , α h ∈ C . Let(12) b [ µ,d,η ] ( t ) := 1( t − t − s ) η ( t ) det p d + h +1 ( t ) p d + h +1 ( s ) p d + h +1 (1) p d + h +1 ( α ) . . . p d + h +1 ( α h )... p d ( t ) p d ( s ) p d (1) p d ( α ) . . . p d ( α h ) p d − ( t ) p d − ( s ) p d − (1) p d − ( α ) . . . p d − ( α h ) , and(13) r [ µ,d,η ] ( t ) := d − X i =0 p i ( t ) det p d + h ( s ) p d + h (1) p d + h ( α ) . . . p d + h ( α h )... p d ( s ) p d (1) p d ( α ) . . . p d ( α h ) p i ( s ) p i (1) p i ( α ) . . . p i ( α h ) . We prove that b [ µ,d,η ] ( t ) and r [ µ,d,η ] ( t ) are scalar multiple of each other. In the special case ofpolynomials we have the following explicit expressions for f [ µ,d,η ] ( t ) . Theorem 1.12.
We have f [ µ,d,η ] ( t ) = cb [ µ,d,η ] ( t ) where c ∈ C . Moreover, f [ µ,d,η ] ( t ) = c ′ r [ µ,d,η ] ( t ) , where c ′ ∈ C . Note that c ′ c = Q d + h +1 i = d a i > . By using the above explicit formula for f [ µ,d,η ] ( t ) , we determine the set of d for which η is ( µ, s, d )-positive in the following theorem. heorem 1.13. η is ( µ, s, d ) -positive if and only if there exists a non-zero κ ∈ C such that forevery ≤ i ≤ d − κ det p d + h ( s ) p d + h (1) p d + h ( α ) . . . p d + h ( α h ) ... p d ( s ) p d (1) p d ( α ) . . . p d ( α h ) p i ( s ) p i (1) p i ( α ) . . . p i ( α h ) ≥ , and for every d − ≤ j ≤ d + h (15) κ det γ d + h +1 ,j p d + h +1 ( s ) p d + h +1 (1) p d + h +1 ( α ) . . . p d + h +1 ( α h ) ... ... γ d,j p d ( s ) p d (1) p d ( α ) . . . p d ( α h ) γ d − ,j p d − ( s ) p d − (1) p d − ( α ) . . . p d − ( α h ) ≥ , where γ i,j = Z − ( p i ( t ) − p i (1))( t − p j ( t ) dµ = R − t − det " p j (1) p i (1) p j ( t ) p i ( t ) dµ for j < i, otherwise. As the value of the functional L , we have the following theorem. Theorem 1.14.
For η ( t ) = ( t − α ) . . . ( t − α h ) real non-negative polynomial with distinct roots α i ∈ C , L ( g [ µ,d,η ] ) = ω det p ′ d + h +1 (1) p d + h +1 ( s ) p d + h +1 (1) p d + h +1 ( α ) . . . p d + h +1 ( α h ) ... p ′ d (1) p d ( s ) p d (1) p d ( α ) . . . p d ( α h ) p ′ d − (1) p d − ( s ) p d − (1) p d − ( α ) . . . p d − ( α h ) det p d + h +1 (1) P d + hl = d − a l +1 p l (1) p l +1 (1) p d + h +1 ( s ) p d + h +1 (1) p d + h +1 ( α ) . . . p d + h +1 ( α h ) ... ... p d (1) P d − l = d − a l +1 p l (1) p l +1 (1) p d ( s ) p d (1) p d ( α ) . . . p d ( α h )0 p d − ( s ) p d − (1) p d − ( α ) . . . p d − ( α h ) . Our framework subsumes the work of Levenshtein, by recovering his extremal polynomials in bothodd and even degrees with η = 1 and η = 1 + t .1.7. Structure of the paper.
This paper is structured as follows. Section 2 gives a summary ofthe properties of orthogonal polynomials, especially Jacobi polynomials, that will be used in thispaper. Except for Proposition 2.1, there is no claim of originality in this section. In Section 3,we setup some of the notation used in this paper and prove Proposition 1.3. Section 4 concernsthe general construction of our test functions that are used in conjunction with the Delsarte andCohn–Elkies linear programming methods; we prove Propositions 4.6 and 4.7 in this section. InSection 5, we prove our main Theorems 1.4 and 1.6. In this section, we use our estimates on thetriple density functions proved in Section 7. We also use our local approximation to Jacobi polyno-mials proved in Section 6. Sections 8, 9, and 10 concern generalizations of Levenshtein’s optimalpolynomials and their properties. In the final Section 11, we provide a table of improvement factors.
Acknowledgments.
Both authors are thankful to Alexander Barg, Peter Boyvalenkov, Henry Cohnand Peter Sarnak for informing us of relevant literature. In particular, we would like to thank enry Cohn for a fruitful email exchange. Both authors would also like to thank Mehrdad KhaniShirkoohi and Hamid Zargar for assistance regarding our code. N.T.Sardari would like to thankMatthew de Courcy-Ireland for enlightening conversations regarding the sphere packing problem,and also the mathematics department of EPFL for their hospitality. N.T.Sardari’s work is sup-ported partially by the National Science Foundation under Grant No. DMS-2015305 N.T.S andis grateful to the Institute for Advanced Study and the Max Planck Institute for Mathematics inBonn for its hospitality and financial support. M.Zargar would also like to thank the Max PlanckInstitute for Mathematics in Bonn for a short visit in December of 2019. M.Zargar is supported bySFB1085 at the University of Regensburg.2. Orthogonal polynomials
In this section, we record some well-known properties of the Jacobi polynomials (see [Sze39, ChapterIV]) as well as the Christoffel-Dabroux formula that will be used repeatedly in this paper, especiallyin the later sections. Except possibly for Proposition 2.1, there is no claim of originality.We denote by p α,βn ( t ) the Jacobi polynomial of degree n with parameters α and β . These are or-thogonal polynomials with respect to the measure dµ α,β := (1 − t ) α (1 + t ) β dt/ R − (1 − t ) α (1 + t ) β dt on the interval [ − , α = β , we denote this measure simply as dµ α . For simplicity, wewrite p n ( t ) for the L -normalized Jacobi polynomials p α,αn ( t ) || p α,αn || . We denote the top coefficient of p n ( t )with k n . Note that the weight ( α, α ) is implicit in the notation. The
Christoffel-Darboux formulastates the following (see [Sze39, Theorem 3.2.2]):(16) 1( t − s ) det (cid:20) p n +1 ( t ) p n +1 ( s ) p n ( t ) p n ( s ) (cid:21) = k n +1 k n n X j =0 p j ( s ) p j ( t ) = a n +1 n X j =0 p j ( s ) p j ( t ) , where a n +1 = k n +1 k n > . In fact, this formula holds more generally for sequences of orthonormalpolynomials with respect to some measure on [ − , p n +1 ( t ) = ( a n +1 t + b n +1 ) p n ( t ) − c n +1 p n − ( t ) , where(18) c n +1 = a n +1 a n = k n +1 k n − k n > , and b n +1 = 0 for α = β. We also have; see [KL78, Equation (38)](19) p i ( t ) p j ( t ) = i + j X l =0 a li,j p l ( t ) , where a li,j ≥ α ≥ β , which means dµ α satisfies the Krein condition. The Jacobi polynomialsthat we use are suitably normalized so that we have the following formulas:(20) p α,βn (1) = (cid:18) n + αn (cid:19) , (21) ω α,βn := Z − ( p α,βn ) dµ α,β ( t ) = 2 α + β +1 Γ( n + α + 1)Γ( n + β + 1)(2 n + α + β + 1)Γ( n + α + β + 1) n ! for α, β > − , n ( n + α + β )(2 n + α + β − p α,βn ( t ) = (2 n + α + β − n + α + β )(2 n + α + β − t + α − β ) p α,βn − ( t ) − n + α − n + β − n + α + β ) p α,βn − ( t ) for n ≥ , (22) nd(23) ddt p α,βn ( t ) = n + α + β + 12 p α +1 ,β +1 n − ( t ) . Henceforth, we suppress the α from ω α,αn and write simply ω n .When proving our local approximation result on Levenshtein’s optimal polynomials, we will use thefact that the Jacobi polynomial p ( α,β ) d ( t ) satisfies the differential equation(24) (1 − t ) x ′′ ( t ) + ( β − α − ( α + β + 2) t ) x ′ ( t ) + d ( d + α + β + 1) x ( t ) = 0 . We also use the following expression for a n appearing in equation (17) above. This is easily obtainedfrom the other properties above.(25) a n = (2 n + α + β ) p (2 n + α + β + 1)(2 n + α + β − p n ( n + α )( n + β )( n + α + β ) . In this paper, we will perform a change of basis of polynomials by changing the measure withrespect to which orthogonality is defined. The following proposition will be useful.
Proposition 2.1.
Suppose that { p , p , . . . } is an orthonormal basis for R [ t ] with respect to themeasure dµ on [ − , . For distinct α , . . . , α k ∈ C , let d ˜ µ ( t ) := ( t − α ) . . . ( t − α k ) dµ ( t ) and ˜ p i ( t ) := 1( t − α ) . . . ( t − α k ) det p i + k ( t ) p i + k ( α ) . . . p i + k ( α k ) p i + k − ( t ) p i + k − ( α ) . . . p i + k − ( α k ) ... ... ... p i ( t ) p i ( α ) . . . p i ( α k ) . Then { ˜ p , ˜ p , . . . } forms an orthogonal family of polynomials for d ˜ µ ( t ) . Proof.
Suppose that n > m ≥ Z − ˜ p m ˜ p n d ˜ µ ( t ) = 0 . We have Z − ˜ p m ˜ p n d ˜ µ ( t ) = Z − ˜ p m det p n + k ( t ) p n + k ( α ) . . . p n + k ( α k ) p n + k − ( t ) p n + k − ( α ) . . . p n + k − ( α k )... ... ... p n ( t ) p n ( α ) . . . p n ( α k ) dµ ( t ) = 0 , where the last identity follows from the fact that ˜ p m is a polynomial of degree m < n which isorthogonal to any linear combination of p n , . . . , p n + k with respect to measure dµ ( t ) . This completesthe proof of our proposition. (cid:3) Geometric improvement
In this section, we give a proof of Proposition 1.3. First, we introduce some notations.Let 0 < θ < θ ′ < π be given angles, and let s := cos θ and s ′ := cos θ ′ as before. Through-out, S n − is the unit sphere. Suppose z ∈ S n − is a fixed point. Consider the cap Cap θ,θ ′ ( z ) on S n − centered at z and of radius √ − r , where r = r s − s ′ − s ′ . onsider the tangent hyperplane T z S n − := { u ∈ R n : h u , z i = 0 } to S n − at the point z ∈ S n − .For each x , y ∈ S n − of angle at least θ from each other, we may orthogonally project them onto thetangent plane T z S n − via the map Π z : S n − → T z S n − . It is easy to see that for every x ∈ S n − ,Π z ( x ) = x − h x , z i z . For brevity, we denote Π z ( x ) | Π z ( x ) | by ˜ x when z is understood. Given x , y ∈ S n − , we obtain points˜ x , ˜ y in the tangent space T z S n − . We will use the following notation. u := h x , z i ,v := h y , z i , and t := h x , y i . It is easy to see that h ˜ x , ˜ y i = t − uv p (1 − u )(1 − v ) . For 0 < θ < θ ′ < π , we define(26) γ θ,θ ′ := 2 arctan s p (1 − s )( s − s ′ ) + arccos( r ) − π, and R := cos( γ θ,θ ′ ) . Then
R > r , and we define the stripStr θ,θ ′ ( z ) := (cid:8) x ∈ S n − : arccos( R ) ≤ h x , z i ≤ arccos( r ) (cid:9) . By equation (147) and Lemma 42 of [AVZ00], the maximum of s − uv p (1 − u )(1 − v )over this strip occurs when x , y lie on the boundary of the cap Cap θ,θ ′ ( z ), and so is at most s − r − r = s ′ . Consequently, any two points in the strip Str θ,θ ′ ( z ) that are at least θ apart are projected underΠ z to points on the tangent plane to z that are at least θ ′ apart from the point of view of z . Withthis in mind, we are now ready to prove Proposition 1.3. When discussing the measure of stripsStr θ,θ ′ ( x ), we drop x from the notation and simply write Str θ,θ ′ . Proof of Proposition 1.3.
Suppose { x , . . . , x N } ⊂ S n − is a maximal spherical code correspondingto the angle θ . Given x ∈ S n − , let m ( x ) be the number of such strips Str θ,θ ′ ( x i ) such that x ∈ Str θ,θ ′ ( x i ). Note that x ∈ Str θ,θ ′ ( x i ) if and only if x i ∈ Str θ,θ ′ ( x ). Therefore, the strip Str θ,θ ′ ( x )contains m ( x ) points of { x , . . . , x N } . From the previous lemma, we know that the projection ofthese m ( x ) points onto the tangent plane of S n − at the points x ∈ S n − have pairwise radialangles at least θ ′ . As a result, m ( x ) ≤ M ( n − , θ ′ ) , using which we obtain N · µ (Str θ,θ ′ ) = N X i =1 Z Str θ,θ ′ ( x i ) dµ ( x ) = Z S n − m ( x ) dµ ( x ) ≤ M ( n − , θ ′ ) Z S n − dµ ( x ) = M ( n − , θ ′ ) . Hence, M ( n, θ ) ≤ M ( n − , θ ′ ) µ (Str θ,θ ′ ) , s required.Note that the masses of Str θ,θ ′ ( x ) and the cap Cap θ,θ ′ have the property that1 − µ (Str θ,θ ′ ) µ n ( θ, θ ′ ) = 1 µ n ( θ, θ ′ ) Z R (1 − t ) n − dt ≤ (1 − R ) n − µ n ( θ, θ ′ ) . On the other hand, we may also give a lower bound on µ n ( θ, θ ′ ) by noting that µ n ( θ, θ ′ ) = 1 ω Z r (1 − t ) n − dt ≥ (cid:16)q r − r (cid:17) n − ω Z r (1 − t ) n − dt = (cid:16)q r − r (cid:17) n − (1 − r ) n − ( n − ω ≥ (1 − r )(1 − r ) n − n − . Here, ω is the volume of the unit sphere S n − . Combining this inequality with the above, weobtain 1 ≥ µ (Str θ,θ ′ ) µ n ( θ, θ ′ ) ≥ − n − − R ) n − (1 − r )(1 − r ) n − = 1 − n − − r ) e − n − log (cid:16) − r − R (cid:17) . Since
R > r , this lower bound exponentially converges to 1 as n → ∞ . (cid:3) New test functions
In this section, we give linear programming bounds on sizes of spherical codes and sphere packingdensities by constructing new test functions using averaging arguments.4.1.
Spherical codes.
In this subsection, we construct a function inside D ( dµ n − , cos θ ) from agiven one inside D ( dµ n − , cos θ ′ ) , where θ ′ > θ. Suppose that g θ ′ ∈ D ( dµ n − , cos θ ′ ) . Fix z ∈ S n − . Given x , y ∈ S n − , we define(27) h ( x , y ; z ) := F ( h x , z i ) F ( h y , z i ) g θ ′ ( h ˜ x , ˜ y i ) , where F is an arbitrary integrable real valued function on [ − , x and ˜ y are are unit vectorson the tangent space of the sphere at z as defined in the previous section. We also use the notation u, v, t as before. It is easy to see that h ( x , y ; z ) := F ( u ) F ( v ) g θ ′ t − uv p (1 − u )(1 − v ) ! . The above types functions in three variables also appear in semi-definite linear programing [BV08].Indeed this was our main motivation for considering these types of functions.
Lemma 4.1. h ( x , y ; z ) is a positive definite function in variables x , y on S n − , namely X x i ,x j ∈ A a i a j h ( x i , x j ; z ) ≥ or every finite subset A ⊂ S n − , and coefficients a i ∈ R . Moreover, h is invariant by the diagonalaction of O(n), namely h ( x , y ; z ) = h ( k x , k y ; k z ) for every k ∈ O ( n ) . Proof.
It follows easily from the definitions. (cid:3)
Let(28) h ( x , y ) := Z O ( n ) h ( x , y ; k z ) dµ ( k ) , where dµ ( k ) is the normalized Haar measure on O ( n ) . Lemma 4.2. h ( x , y ) is a positive definite point pair invariant function on S n − Proof.
It follows form the previous lemma. (cid:3)
Since h ( x , y ) is a point pair invariant function, so it only depends on t = h x , y i . For the rest of thispaper, we abuse notation and consider h as a real valued function on [ − , , where h ( x , y ) = h ( t ) . Computing L ( h ) . We now proceed to computing the value of L ( h ) in terms of F and g θ ′ . First, we compute the value of h (1) . Let k F k := R − F ( u ) dµ n − ( u ) . Lemma 4.3.
We have h (1) = g θ ′ (1) k F k . Proof.
Indeed, by definition, h (1) corresponds to taking x = y , from which it follows that t = 1, u = v and t − uv √ (1 − u )(1 − v ) = 1. Therefore, we obtain h (1) = Z − F ( u ) g θ ′ (1) dµ n − ( u ) = g θ ′ (1) k F k . (cid:3) Next, we compute the zero Fourier coefficient of h. Let F = R − F ( u ) dµ n − ( u ) and g θ ′ , = R − g θ ′ dµ n − ( u ). Lemma 4.4.
We have h = g θ ′ , F . Proof.
Let O ( n − ⊂ O ( n ) be the centralizer of z . We identify O ( n ) /O ( n −
1) by S n − via themap [ k ] := k z ∈ S n − . Then we write the Haar measure of O ( n ) as the product of the Haarmeasure of O ( n −
1) and surface area dσ of S n − dµ ( k ) = dµ ( k ′ ) dσ ([ k ]) . where k ′ ∈ O ( n − . By equation (27), (28) and the above, we obtain h = Z k ′ i ∈ O ( n − Z [ k i ] ∈ S n − F ( (cid:10) k ′ [ k ] , z (cid:11) ) F ( (cid:10) k ′ [ k ] , z (cid:11) ) g θ ′ (cid:16)D ^ k ′ [ k ] , ^ k ′ [ k ] E(cid:17) dµ ( k ′ ) dσ ([ k ]) dµ ( k ′ ) dσ ([ k ])= Z [ k i ] ∈ S n − F ( h [ k ] , z i ) F ( h [ k ] , z i ) dσ ([ k ]) dσ ([ k ]) Z k ′ i ∈ O ( n − g θ ′ (cid:16)D k ′ g [ k ] , k ′ g [ k ] E(cid:17) dµ ( k ′ ) dµ ( k ′ ) . We note that Z k ′ i ∈ O ( n − g θ ′ (cid:16)D k ′ g [ k ] , k ′ g [ k ] E(cid:17) dµ ( k ′ ) dµ ( k ′ ) = g θ ′ , , nd Z [ k i ] ∈ S n − F ( h [ k ] , z i ) F ( h [ k ] , z i ) dσ ([ k ]) dσ ([ k ]) = Z − F ( u ) F ( v ) dµ n − ( u ) µ n − ( v ) = F . Therefore, h = g θ ′ , F , as required. (cid:3) Proposition 4.5.
We have L ( h ) = L ( g θ ′ ) k F k F . Proof.
This follows immediately from Lemma 4.3 and Lemma 4.4. (cid:3)
Criteria for h ∈ D ( dµ n − , cos θ ) . Finally, we give a criterion which implies h ∈ D ( dµ n − , cos θ ) . Recall that 0 < θ < θ ′ , and 0 < r < R < s ′ = cos( θ ′ ) and s = cos( θ ) . Note that s ′ < s , r = q s − s ′ − s ′ , s ′ = s − r − r and R = cos γ θ,θ ′ as in equation (26). We define χ ( y ) := ( r < y ≤ R, . Proposition 4.6.
Suppose that g θ ′ ∈ D ( dµ n − , cos θ ′ ) is given and h is defined as in (28) for some F . Suppose that F ( x ) is a positive integrable function giving rise to an h such that h ( t ) ≤ forevery − ≤ t ≤ cos θ . Then h ∈ D ( dµ n − , cos θ ) , and M ( n, θ ) ≤ L ( h ) = L ( g θ ′ ) k F k F . In particular, among all positive integrable functions F with compact support inside [ r, R ] , χ mini-mize the value of L ( h ) , and for F = χ we have M ( n, θ ) ≤ L ( h ) ≤ L ( g θ ′ ) µ n ( θ, θ ′ ) (1 + O ( ne − nc )) , where c = log (cid:16) − r − R (cid:17) > . Proof.
The first part follows from the previous lemmas and propositions. Let us specialize to thesituation where F is merely assumed to be a positive integrable function with compact supportinside [ r, R ]. Let us first show that h ( t ) ≤ t ≤ s . We have h ( t ) := Z O ( n ) h ( x , y ; k z ) dµ ( k ) , where h ( x , y ; z ) := F ( u ) F ( v ) g θ ′ (cid:18) t − uv √ (1 − u )(1 − v ) (cid:19) . First, note that F ( u ) F ( v ) = 0 precisely when x and y belong to Str θ,θ ′ ( z ) . By equation (147) and Lemma 42 of [AVZ00], the angular distancebetween the projected vectors is at least θ ′ , and so t − uv p (1 − u )(1 − v ) ∈ [ − , cos θ ′ ] . Therefore g θ ′ t − uv p (1 − u )(1 − v ) ! ≤ , hen F ( u ) F ( v ) = 0 . Since the integrand h ( x , y , z ) is non-positive when t ∈ [ − , cos θ ]. Conse-quently, h ( t ) ≤ t < s .It is easy to see that when F = χ , then k F k = µ (Str θ,θ ′ ( z ))and F = µ (Str θ,θ ′ ( z )) . Therefore, by our estimate in the proof of Proposition 1.3 we have M ( n, θ ) ≤ L ( h ) ≤ L ( g θ ′ ) µ n ( θ, θ ′ ) (cid:18) O ( ne − n log (cid:16) − r − R (cid:17) ) (cid:19) . Finally, the optimality follows from the Cauchy–Schwarz inequality. More precisely, since F ( x ) hascompact support inside [ r, R ], we have µ (Str θ,θ ′ ( z )) k F k ≥ F . Therefore, L ( h ) = g θ ′ (1) g θ ′ , k F k F ≥ L ( g θ ′ ) µ (Str θ,θ ′ ( z )) with equality only when F = χ. (cid:3) Remark . Proposition 4.6 shows that χ is optimal among all functions F with support [ r, R ] . Weused this restriction on the support of F in order to prove that h ( t ) ≤ t < s. By continuity,this negativity condition will continue to hold if we expand slightly the support of χ to [ r − δ, R ] forsome δ > . In order to determine explicitly the extent to which we can enlarge the support of χ ,we should understand the behaviour of g θ ′ at its zero cos θ ′ . As Levenshtein’s function g θ ∗ may bewritten in terms of Jacobi polynomials, this may be understood by understanding the behaviourof Jacobi polynomials at their extreme roots as the dimension and degree grow. We will make thisprecise in the next section.4.2. Sphere packings.
Suppose 0 < θ ≤ π is a given angle, and suppose g θ ∈ D ( dµ n − , cos θ ).Fixing z ∈ R n , for each x , y ∈ R n consider H ( x , y ; z ) := F ( | x − z | ) F ( | y − z | ) g θ ( h ˜ x , ˜ y i ) , where F is a positive function on R such that as a radial function it is in L ( R n ) ∩ L ( R n ). The tildenotation denotes normalization to a unit vector from z . We may then define H ( x , y ) by averagingover all z ∈ R n : H ( x , y ) := Z R n H ( x , y ; z ) d z . Note that this then makes H ( x , y ) into a point-pair invariant function. As before, we abuse notationand write H ( T ) instead of H ( x , y ) when T = | x − y | . The analogue of Proposition 4.6 is then thefollowing. Proposition 4.7.
Let < θ ≤ π and suppose g θ ∈ D ( dµ n − , cos θ ) . Suppose F is as above suchthat H ( T ) ≤ for every T ≥ . Then H is positive definite on R n and (30) δ n ≤ vol( B n ) k F k L ( R n ) n k F k L ( R n ) L ( g θ ) . In particular, if F = χ [0 ,r + δ ] , where δ ≥ and r + δ ≤ , is such that it gives rise to an H satisfying H ( T ) ≤ for every T ≥ , then (31) δ n ≤ L ( g θ )(2( r + δ )) n . roof. The proof of this proposition is similar to that of Theorem 3.4 of Cohn–Zhao [CZ14]. Thepositive-definiteness of H follows as before. We focus our attention on proving inequality (30).Suppose we have a packing of R n of density ∆ by non-overlapping balls of radius . By Theorem3.1 of Cohn–Elkies [CE03], we have ∆ ≤ vol( B n ) H (0)2 n b H (0) . Note that H (0) = g θ (1) k F k L ( R n ) , and that b H (0) = Z R n H ( | z | ) d z = g θ, k F k L ( R n ) . As a result, we obtain inequality (30). The rest follows from a simple computation. (cid:3)
Note that the situation δ = 0 and r =
12 sin( θ/ with π/ ≤ θ ≤ π corresponds to Theorem 3.4 ofCohn–Zhao [CZ14], as checking the negativity condition H ( T ) ≤ T ≥ n comes from considering functions where the negativity condition isfor T ≥ T ≥ Remark . In this paper, we primarily consider characteristic functions; however, it is an interest-ing open question to determine the optimal such F in order to obtain the best bounds on spherepacking densities through this method.5. Comparison with previous bounds
In this section, we begin by improving upon the Levenshtein bound on θ -spherical codes. In thesecond subsection, we improve upon the Cohn–Zhao upper bound on sphere packing densities. Notethat the constructions of the functions were provided in the previous Section 4. We complete theproof of our main theorems by providing the necessary functions F for applying Propositions 4.6and 4.7.5.1. Improving spherical codes bound.
Recall Theorem 1.1. In this section, we give a proofof Theorem 1.4].
Proof Theorem 1.4.
Recall that 0 < θ < θ ′ ≤ π/
2. Let s ′ = cos( θ ′ ) = t α +1 ,α + ε ,d − ε for some ε ∈ { , } and α := n − , and s = cos( θ ) . Note that s ′ < s and for r = q s − s ′ − s ′ , we have s ′ = s − r − r and0 < r <
1. Let 0 < δ = O ( n ) that we specify later, and define the function F for the applicationof Proposition 4.6 to be F ( y ) := χ ( y ) := ( r − δ < y ≤ R, . We cite some results from the work of Levenshtein [Lev98]. By [Lev98, Lemma 5.89] t α +1 ,α +11 ,d − < t α +1 ,α ,d < t α +1 ,α +11 ,d . Levenshtein proved Theorem 1.1, by using the following test functions; see [Lev98, Lemma 5.38] g ( x ) = ( x +1) ( x − s ′ ) (cid:16) p α +1 ,α +1 d − ( x ) (cid:17) if s ′ = t α +1 ,α +11 ,d − , ( x +1)( x − s ′ ) (cid:16) p α +1 ,αd ( x ) (cid:17) if s ′ = t α +1 ,α ,d , where α := n − in our case. Applying Proposition 4.6 with the function F as above and the func-tion g , we obtain the inequality in the statement of the theorem. We now prove the bound on c n for sufficiently large n . y Corollary 1.5, we know that if s ′ = cos( θ ∗ ) as n → ∞ , then L ( g θ ′ ) /µ n ( θ, θ ′ ) for large n isexponentially worse than L ( g θ ∗ ) /µ n ( θ, θ ∗ ). Therefore, we assume that s ′ = cos( θ ′ ) = t α +1 ,α + ε ,d − ε forsome ε ∈ { , } , and that s ′ is sufficiently close to cos θ ∗ . Note that Corollary 1.5 only used thefirst part of Theorem 1.4 (basically, Proposition 4.6), and so our argument is not circular. Suppose s ′ = t α +1 ,α ,d . By Proposition 6.1, we have g ( x ) = ( x + 1)( x − s ′ ) dp ( α +1 ,α ) d dt ( s ′ ) (1 + A ( x )) , where | A ( x ) | ≤ e σ ( x ) − σ ( x ) − σ ( x ) := | x − s ′ | ( ns ′ + s ′ + 1)1 − s ′ . Next, we consider the test functions constructed in the previous section, and check its negativity.By Proposition 7.1, we have µ ( x ; s, χ ) = (1 + o (1)) 2 √ (cid:18) δ + r s − x − x − r (cid:19) + (cid:18)(cid:18) − x x (cid:19) (cid:0) s − r (cid:1) (cid:19) n − e − nr (cid:18) √ s − x − x − r (cid:19) s − r . We need to find the maximal δ > Z − g ( x ) µ ( x ; t, χ ) dx ≤ − ≤ t ≤ s . We note that it is enough to prove the above inequality for t = s. Since for s ′ < t < s , r ( t ) = r t − s ′ − s ′ < r s − s ′ − s ′ − O (1 /n ) < r − δ. Let δ = O (1 /n ) that we specify later. Note that the integrand is negative for x < s ′ and positivefor x > s ′ . Hence, sign (cid:18)Z − g ( x ) µ ( x ; s, χ ) dx (cid:19) = sign (cid:12)(cid:12)(cid:12)(cid:12)Z s ′ g ( x ) µ ( x ; s, χ ) dx (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z s ′ − g ( x ) µ ( x ; s, χ ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! Next, we give a lower bound on the absolute value of the integral over − < x < s ′ . By our upperbound on A ( x ), we have | A ( x ) | ≥ − e σ ( x ) − σ ( x ) ! + We note that 2 − e σ − σ is a concave function with value 1 at σ = 0 and a root at σ = 1 . ... .Hence, for σ < . | A ( x ) | ≥ − e σ ( x ) − σ ( x ) ! . Note that σ ( x ) < . | x − s ′ | ≤ (1 . − s ′ ) ns ′ . et λ := (1 . − s ′ ) ns ′ . Therefore, as n → ∞ − (cid:0) s − r (cid:1) − ( n − dp ( α +1 ,α ) d dt ( s ′ ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z s ′ − g ( x ) µ ( x ; s, χ ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (33) & Z s ′ s ′ − λ (1 + x )( s ′ − x ) − e σ ( x ) − σ ( x ) ! (cid:18) δ + r s − x − x − r (cid:19) (cid:18) − x x (cid:19) n − e − nr (cid:18) √ s − x − x − r (cid:19) s − r dx. We change the variable s ′ − x to z and note that(34) r s − x − x − r = z (1 − s )2( s − s ′ ) / (1 − s ′ ) / + O ( λ )(35) 1 − x x = 1 − s ′ s ′ (cid:18) zs ′ (1 − s ′ ) + O ( λ ) (cid:19) for | x − s ′ | < λ. Hence, we obtain that as n → ∞ and | x − s ′ | < λ , e − nr (cid:18) √ s − x − x − r (cid:19) s − r = e − nr z (1 − s )2( s − s ′ )1 / − s ′ )3 / ! s − r + O ( λ ) = e (cid:16) − nzs ′ (1 − s ′ ) (cid:17) + O ( λ ) (cid:18) − x x (cid:19) n − = (cid:18) − s ′ s ′ (cid:19) n − e (cid:16) nzs ′ (1 − s ′ (cid:17) + O ( λ ) − e σ ( x ) − σ ( x ) = − e nzs ′ (1 − s ′ − nzs ′ (1 − s ′ ) (1 + O ( λ )) . for | x − s ′ | < λ. We replace the above asymptotic formulas and obtain that as n → ∞ , the righthand side of (33) is at least(1 + s ′ ) (cid:18) − s ′ s ′ (cid:19) n − Z λ z − e nzs ′ (1 − s ′ − nzs ′ (1 − s ′ ) (cid:18) δ + z (1 − s )2( s − s ′ ) / (1 − s ′ ) / (cid:19) e (cid:16) − nz − s ′ (cid:17) dz. Finally, we give an upper bound on the absolute value of the integral over s ′ < x < . We note that r s − x − x − r = z (1 − s )2( s − s ′ ) / (1 − s ′ ) / + O ( λ ) . Let Λ := s − s ′ ) / (1 − s ′ ) / δ (1 − s ) . Hence, we have (cid:18) δ + r s − x − x − r (cid:19) + = 0for x − s ′ > s − s ′ ) / (1 − s ′ ) / δ (1 − s ) . We have | A ( x ) | ≤ e σ ( x ) − σ ( x ) ! . here σ ( x ) = n | x − s ′ | s ′ (1 − s ′ ) . Therefore,2 − (cid:0) s − r (cid:1) − ( n − dp ( α +1 ,α ) d dt ( s ′ ) − (cid:12)(cid:12)(cid:12)(cid:12)Z s ′ g ( x ) µ ( x ; s, χ ) dx (cid:12)(cid:12)(cid:12)(cid:12) . (1 + s ′ ) (cid:18) − s ′ s ′ (cid:19) n − Z Λ0 z e nzs ′ (1 − s ′ − nzs ′ (1 − s ′ ) (cid:18) δ − z (1 − s )2( s − s ′ ) / (1 − s ′ ) / (cid:19) e (cid:16) nz − s ′ (cid:17) dz. We choose δ such that Z λ z − e nzs ′ (1 − s ′ − nzs ′ (1 − s ′ ) (cid:18) δ + z (1 − s )2( s − s ′ ) / (1 − s ′ ) / (cid:19) e (cid:16) − nz − s ′ (cid:17) dz ≥ Z Λ0 z e nzs ′ (1 − s ′ − nzs ′ (1 − s ′ ) (cid:18) δ − z (1 − s )2( s − s ′ ) / (1 − s ′ ) / (cid:19) e (cid:16) nz − s ′ (cid:17) dz. Since s ′ converges to cos( θ ∗ ) as n → ∞ , for large enough n we may replace the numerical valuecos(1 . s ′ . Furthermore, write v := nz , and divide by δ to obtain Z . v (cid:18) − e . v − . v (cid:19) (cid:16) vn Λ (cid:17) e ( − . v ) dv ≥ Z n Λ0 v (cid:18) e . v − . v (cid:19) (cid:16) − vn Λ (cid:17) e (1 . v ) dv. Here, we have also used that Λ = δ ( s − s ′ ) / (1 − s ′ ) / (1 − s ) . Also, note that nλ = 2 . ... when s ′ isnear cos(1 . v = 0 v (cid:18) e . v − . v (cid:19) e (1 . v ) = v + 1 . v + 1 . v + 1 . v + 0 . v + 0 . v + 0 . v + 0 . v + 0 . v + 0 . v + 0 . v + 0 . v + Er with error | Er | < × − if n Λ < .
92, which we assume to be the case. Simplifying, we want tofind the maximal n Λ such that0 . . n Λ ≥ Z n Λ0 v (cid:18) e . v − . v (cid:19) (cid:16) − vn Λ (cid:17) e (1 . v ) dv = 3 . × − ( n Λ) ( n Λ) + 5 . n Λ) + 30 . n Λ) + 148 . n Λ) + 654 . n Λ) + 2585 . n Λ) + 9002 . n Λ) + 27010 . n Λ) + 67600 . n Λ) + 134116( n Λ) + 192140( n Λ) + 157353 ! − n Λ (cid:16) − . . e . n Λ) − . e . n Λ) + 1 . e . n Λ) (cid:17) + Er, here the error Er again satisfies | Er | ≤ × − . A numerical computation gives us that theinequality is satisfied when n Λ ≤ . ... . Therefore, if we choose δ = ℓ/n , then we must have ℓ ≤ . ... (1 − s )2( s − s ′ ) / (1 − s ′ ) / . In this case, the cap of radius √ − r becomes p − ( r − δ ) = √ − r (cid:16) ℓrn (1 − r ) (cid:17) + O (1 /n ).Note that r = q s − s ′ − s ′ , and so r − r = ( s − s ′ ) / (1 − s ′ ) / − s . We deduce that, ℓr (1 − r ) ≤ . ... (1 − s )2( s − s ′ ) / (1 − s ′ ) / · ( s − s ′ ) / (1 − s ′ ) / − s = 0 . ... − s ′ = 0 . ... Similarly, one may obtain the same when s ′ = t α +1 ,α +11 ,d − . Therefore, our improvement to Leven-shtein’s bound on M ( n, θ ) for large n is by a factor of 1 /e . ... = 0 . ... for any choice ofangle 0 < θ < θ ∗ . As the error is less than 2 × − , we deduce that we have an improvement by afactor of at most 0 . n . (cid:3) Comparison with the Cohn–Zhao bound.
In this subsection, we give our improvementto Cohn and Zhao’s [CZ14] bound on sphere packings. We are in the situation of wanting tobound from above the maximal density δ n − introduced in the introduction to this paper. We have s ′ = cos( θ ′ ), and r = √ − s ′ ) . By assumption, 1 / ≤ s ′ < /
2. In this case, we define for each0 < δ = c n the function F to be used in Proposition 4.7 to be F ( y ) := χ ( y ) := ( ≤ y ≤ r + δ, . Let r := p − (1 − x )( r + δ ) + x ( r + δ ) . Suppose that | x − s ′ | ≤ c n and δ = c n for some 0 < c < .
81, 0 < c < .
36. Note that for sucha c and n ≥ s ′ < / r + δ <
1. By Proposition 7.2, we have µ ( x ; χ ) = 0 for r ≥ r + δ. Otherwise, µ ( x ; χ ) = C n (1 + E )(1 − x ) n − (cid:18) − x ) (cid:19) n − vuut x − (1 − x ) r p − (1 − x ) r ! ( r + δ − r )for some constant C n > µ ( x ; χ ) a probability measure on [ − , | E | < (4 c +2 c +2) n for n ≥ c , c , we have | E | < n . Proof of Theorem 1.6.
As in the proof of Theorem 1.4, we consider g ( x ) chosen for s ′ = cos( θ ′ ).Note that Proposition 4.7 applied to the function F above and g Levenshtein’s optimal polynomialfor angle θ ′ gives the existence of the inequality in the theorem. We now deal with the more preciseversion when n ≥ s ′ is a root of the Jacobi polynomial as before. As in the proof ofTheorem 1.4, we let s ′ = t α +1 ,α ,d and take g for this s ′ as before. As before, we begin with theobservation thatsign (cid:18)Z − g ( x ) µ ( x ; χ ) dx (cid:19) = sign (cid:12)(cid:12)(cid:12)(cid:12)Z s ′ g ( x ) µ ( x ; χ ) dx (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z s ′ − g ( x ) µ ( x ; χ ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! . et us give a lower bound on the absolute value of the negative contribution from x ≤ s ′ . As before,for − ≤ x ≤ s ′ , we have | A ( x ) | ≥ − e σ ( x ) − σ ( x ) ! , where σ ( x ) = | x − s ′ | (( n − s ′ + 1)(1 − s ′ ) . Note that σ ( x ) < . A ( x ) is non-negative,implies | x − s ′ | ≤ . − s ′ )( n − s ′ + 1 . Let λ := 1 . − s ′ )( n − s ′ + 1 . Since 1 / ≤ s ′ < / n ≥ λ is compatible with the restriction 0 < c < .
36. Fromthe above considerations, we have C − n n − dp ( α +1 ,α ) d dt ( s ′ ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z s ′ − g ( x ) µ ( x ; χ ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − n n − dp ( α +1 ,α ) d dt ( s ′ ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z s ′ s ′ − λ g ( x ) µ ( x ; χ ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (1 + E ) Z s ′ s ′ − λ (1 + x )( s ′ − x )(1 + A ( x )) (1 − x ) n − (1 − x ) n − vuut x − (1 − x ) r p − (1 − x ) r ! ( r + δ − r ) + dx ≥ (cid:18) − n (cid:19) Z s ′ s ′ − λ ( s ′ − x ) − e σ ( x ) − σ ( x ) ! (cid:18) x − x (cid:19) n − vuut x − (1 − x ) r p − (1 − x ) r ! ( r + δ − r ) dx. Note that if we let z := s ′ − x , then (cid:18) x − x (cid:19) n − = (cid:18) s ′ − s ′ (cid:19) n − e − ( n − z − s ′ (1 + E ′ )where | E ′ | ≤ ( s ′ + λ ) n (1 − ( s ′ + λ ) ) (cid:18) . − s ′ ) s ′ (cid:19) + . n ≤ . n for | x − s ′ | ≤ λ and n ≥ / ≤ s ′ < / x = s ′ , we obtain vuut x − (1 − x ) r p − (1 − x ) r ! = √ E ′′ ) , where | E ′′ | < √ . − s ′ )2 ns ′ ≤ . n for | x − s ′ | ≤ λ , n ≥ / ≤ s ′ < / r in two variables δ and z at δ = z = 0 andobtain r + δ − r = 2 δ + r − s ′ z + O ( δ | z | ) . hen n ≥ n . − s ′ ) s ′ + 0 . ! ≤ . n . As a result, we obtain C − n n − − / (cid:18) s ′ − s ′ (cid:19) − n − dp ( α +1 ,α ) d dt ( s ′ ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z s ′ − g ( x ) µ ( x ; χ ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (1 − E ′′′ ) Z λ z − e (( n − s ′ +1) z (1 − s ′ − (( n − s ′ +1) z (1 − s ′ ) e − ( n − z − s ′ (2 δ + r − s ′ z − . n ) dz ≥ (1 − E ′′′ ) Z λ z − e (( n − s ′ +1) z (1 − s ′ − (( n − s ′ +1) z (1 − s ′ ) e − (( n − s ′ +1) zs ′ (1 − s ′ (2 δ + r − s ′ z − . n ) dz, where 0 < E ′′′ < /n . Making the substitution v = (( n − s ′ +1) z (1 − s ′ ) and dividing by 2 δ , we obtainthat C − n (cid:18) (( n − s ′ + 1)(1 − s ′ ) (cid:19) δ − n − − / (cid:18) s ′ − s ′ (cid:19) − n − dp ( α +1 ,α ) d dt ( s ′ ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z s ′ − g ( x ) µ ( x ; χ ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (1 − E ′′′ ) 12 δ Z . v (cid:18) − e v − v (cid:19) e − vs ′ (cid:18) δ + r (1 + s ′ ) v ( n − s ′ + 1 − . n (cid:19) dv ≥ (cid:18) − n (cid:19) Z . v (cid:18) − e v − v (cid:19) e − vs ′ (cid:18) vβ ( n, s ′ , c ) − . δn (cid:19) dv where β ( n, s ′ , c ) := (( n − s ′ +1)Λ1 − s ′ with Λ := − s ′ ) δr .On the other hand, the integral from s ′ to 1 may be bounded from above in a similar way. Indeed,we may show in a similar way that C − n (cid:18) (( n − s ′ + 1)(1 − s ′ ) (cid:19) δ − n − − / (cid:18) s ′ − s ′ (cid:19) − n − dp ( α +1 ,α ) d dt ( s ′ ) − (cid:12)(cid:12)(cid:12)(cid:12)Z s ′ g ( x ) µ ( x ; χ ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) n (cid:19) Z β ( n,s ′ ,c )0 v (cid:18) e v − v (cid:19) e vs ′ (cid:18) − vβ ( n, s ′ , c ) + 34 . δn (cid:19) dv Note that asymptotically, that is, as n → ∞ , the negativity condition of Proposition 4.7 is satisfiedif(36) Z . v (cid:18) − e v − v (cid:19) e − vs ′ (cid:18) vβ ( ∞ , s ′ , c ) (cid:19) dv ≥ Z β ( ∞ ,s ′ ,c )0 v (cid:18) e v − v (cid:19) e vs ′ (cid:18) − vβ ( ∞ , s ′ , c ) (cid:19) dv. Here, β ( ∞ , s ′ , c ) := lim n →∞ β ( n, s ′ , c ) = c s ′ r (1+ s ′ ) . Note that β ( ∞ , s ′ , c ) is an increasing functionof s ′ in the interval [0 , √ / − / / ≤ s ′ < /
2, we have 0 < β ( ∞ , s ′ , c ) ≤ c . As aresult, for 1 / ≤ s ′ < / Z . v (cid:18) − e v − v (cid:19) e − v (cid:18) v c (cid:19) dv ≥ Z c v (cid:18) e v − v (cid:19) e v (cid:18) − v c (cid:19) dv. y a numerics similar to that done for spherical codes, one finds that the maximal such c < . ... . Therefore, for every 1 / ≤ s ′ < / n → ∞ , we have an improvement at leastas good as e − . /r = exp( − . p − s ′ )) . Note that for such s ′ , as n → ∞ , this gives us an improvement of at most 0 . n ≥ / ≤ s ′ < /
2, given such an n we need tomaximize c < .
81 such that (cid:18) − n (cid:19) Z . v (cid:18) − e v − v (cid:19) e − vs ′ (cid:18) vβ ( n, s ′ , c ) − . δn (cid:19) dv ≥ (cid:18) n (cid:19) Z β ( n,s ′ ,c )0 v (cid:18) e v − v (cid:19) e vs ′ (cid:18) − vβ ( n, s ′ , c ) + 17 . δn (cid:19) dv. Just as in the asymptotic case above, for 1 / ≤ s ′ < / n ≥ β ( n, s ′ , c ) ≤ (cid:16) s ′ + − s ′ (cid:17) c p − s ′ )1 + s ′ ≤ . c , it suffices to find the largest c such that (cid:18) − n (cid:19) Z . v (cid:18) − e v − v (cid:19) e − v (cid:18) . vc − . c n (cid:19) dv ≥ (cid:18) n (cid:19) Z . c v (cid:18) e v − v (cid:19) e v (cid:18) − . vc + 17 . c n (cid:19) dv. This is equivalent to (cid:18) − n (cid:19) (cid:18) . (cid:18) − . c n (cid:19) + 0 . c (cid:19) ≥ (cid:18) n (cid:19) Z . c v (cid:18) e v − v (cid:19) e v (cid:18) − . vc + 17 . c n (cid:19) dv. By a numerical calculation with
Sage , we obtain that the improvement factor for any 1 / ≤ s ′ < / n ≥ .
515 + 74 /n.
On the other hand, if we fix s ′ = s ∗ , then the same kind of calculations as above give us anasymptotic improvement constant of 0 . n ≥ . /n over the combination of Cohn–Zhao [CZ14] and Levenshtein’s optimal polynomials [Lev79a]. In par-ticular, the universal improvement factor 0 . /n for 1 / ≤ s ′ < / n ≥ s ′ = t α +1 ,α +11 ,d − follows in exactly the same way. This completes the proof of ourtheorem. (cid:3) Remark . We end this section by saying that our improvements above are based on a local understanding of Levenshtein’s optimal polynomials, and that there is a loss in our computations.By doing numerics, we may do computations without having to rely on this local understanding.As we will see in Section 11, our numerical calculations lead to even further improvements, even inlow dimensions. . Local approximation of Jacobi polynomials
As part of our proofs in the previous section, we needed to determine local approximations to Jacobipolynomials p ( α,β ) d in the neighbourhood of points s ∈ ( − ,
1) such that s ≥ t α,β ,d , where t α,β ,d denotesthe largest root of p ( α,β ) d . This is obtained using the behaviour of the zeros of Jacobi polynomials.Using this, we obtain suitable local approximations of Levenshtein’s optimal functions near s . Proposition 6.1.
Suppose that α ≥ β ≥ , | α − β | ≤ , d ≥ and s ∈ [ t ( α,β )1 ,d , . Then, we have p ( α,β ) d ( t ) = p ( α,β ) d ( s ) + ( t − s ) dp ( α,β ) d dt ( s )(1 + A ( t )) , where, | A ( t ) | ≤ e σ ( t ) − σ ( t ) − with σ ( t ) := | t − s | (2 αs + 2 s + 1)1 − s . Proof.
Consider the Taylor expansion p ( α,β ) d ( t ) = ∞ X k =0 ( t − s ) k k ! d k p ( α,β ) d dt k ( s )of p ( α,β ) d centered at s . We prove the proposition by showing that for s ∈ [ t ( α,β )1 ,d , k ≥
1. Then we have( d k +1 /dt k +1 ) p ( α,β ) d ( s )( d k /dt k ) p ( α,β ) d ( s ) = ( d/dt ) p ( α + k,β + k ) d − k ( s ) p ( α + k,β + k ) d − k ( s ) = d − k X i =1 s − t α + k,β + ki,d − k ≤ d − X i =1 s − t α +1 ,β +1 i,d − , where the last inequality follows from the fact that the roots of a Jacobi polynomial interlace withthose of its derivative. However, the last quantity is equal to ( d /dt ) p ( α,β ) d ( s )( d/dt ) p ( α,β ) d ( s ) . We proceed to showingthat(38) ( d /dt ) p ( α,β ) d ( s )( d/dt ) p ( α,β ) d ( s ) ≤ α + 2 s + 11 − s . Indeed, we know from the differential equation (24) that(39) (1 − s )( d /dt ) p ( α,β ) d ( s ) + ( β − α − ( α + β + 2) s )( d/dt ) p ( α,β ) d ( s ) + d ( d + α + β + 1) p ( α,β ) d ( s ) = 0 . However, since s is to the right of the largest root of p ( α,β ) d , p ( α,β ) d ( s ) ≥
0. Therefore,(1 − s )( d /dt ) p ( α,β ) d ( s ) + ( β − α − ( α + β + 2) s )( d/dt ) p ( α,β ) d ( s ) ≤ , from which the inequality (38) follows. As a result, the degree k + 1 term compares to the linearterm as ( d k +1 /dt k +1 ) p ( α,β ) d ( s )( d/dt ) p ( α,β ) d ( s ) | t − s | k ( k + 1)! ≤ | t − s | k ( k + 1)! (cid:18) α + 1) s + 11 − s (cid:19) k Consequently, for every k ≥ | t − s | k +1 ( k + 1)! ( d k +1 /dt k +1 ) p ( α,β ) d ( s ) ≤ | t − s | ( d/dt ) p ( α,β ) d ( s ) (cid:16) | t − s | (2 α +2 s +1)1 − s (cid:17) k ( k + 1)! s a result, we obtain that p ( α,β ) d ( t ) = p ( α,β ) d ( s ) + ( t − s ) dp ( α,β ) d dt ( s )(1 + A ( t )) , where | A ( t ) | ≤ e σ ( t ) − σ ( t ) − σ ( t ) = | t − s | (2 αs + 2 s + 1)1 − s . (cid:3) Conditional density functions
In this section, we compute the conditional density functions for spherical codes and sphere packingsused previously.7.1.
Conditional density function for spherical codes.
Let x , y and z be three randomlyindependently chosen points on S n − with respect to the Haar measure. Recall the definitionsof u, v, t previously introduced as the pairwise inner products of these vectors. The pushforwardmeasure onto ( u, v, t ) has the following density function µ ( u, v, t ) = det t ut vu v n − . Let x := t − uv p (1 − u )(1 − v ) . Recall that θ < θ ′ . Let s ′ = cos( θ ′ ) and s = cos( θ ) . Note that s ′ < s and for r = q s − s ′ − s ′ , we have s ′ = s − r − r and 0 < r <
1. Let 0 < δ = o ( √ n ) that we specify later, and define χ ( y ) := ( r − δ < y ≤ R, . Let µ ( x ; s, χ ) be the induced density function on x subjected to the conditions t = s, and r − δ ≤ u, v ≤ R . We define for complex x x + = ( x for x ≥ , Proposition 7.1.
Suppose that | x − s ′ | = o ( √ n ) . We have µ ( x ; s, χ ) = (1 + o (1)) 2 √ (cid:18) δ + r s − x − x − r (cid:19) + (cid:18)(cid:18) − x x (cid:19) (cid:0) s − r (cid:1) (cid:19) n − e − nr (cid:18) √ s − x − x − r (cid:19) s − r . Proof.
We have µ ( x ; s, χ ) = Z C x χ ( u ) χ ( v ) µ ( u, v, s ) dl where the integral is over the curve C x ⊂ R that is given by s − uv √ (1 − u )(1 − v ) = x and dl is theinduced Euclidean metric on C x . We note that µ ( u, v, s ) = (cid:0) (1 − x )(1 − u )(1 − v ) (cid:1) n − . ence, µ ( x ; s, χ ) = Z C x χ ( u ) χ ( v ) µ ( u, v, s ) dl = (1 − x ) n − Z C x χ ( u ) χ ( v ) (cid:0) (1 − u )(1 − v ) (cid:1) n − dl = (1 − x ) n − Z C x χ ( u ) χ ( v ) (cid:18) s − uvx (cid:19) n − dl = (cid:18) − x x (cid:19) n − Z C x χ ( u ) χ ( v ) ( s − uv ) n − dl Suppose that u and v are in the support of χ. Since | x − s ′ | = o ( √ n ) , it follows that u = r + ˜ u and v = r + ˜ v, where ˜ u, ˜ v = o ( √ n ) . We have s − uv = s − r − r (˜ u + ˜ v ) − ˜ u ˜ v = ( s − r ) (cid:18) − r (˜ u + ˜ v ) + ˜ u ˜ vs − r (cid:19) . Hence, µ ( x ; s, χ ) = (cid:18)(cid:18) − x x (cid:19) (cid:0) s − r (cid:1) (cid:19) n − Z C x χ ( u ) χ ( v ) (cid:18) − r (˜ u + ˜ v ) + ˜ u ˜ vs − r (cid:19) n − dl. Recall the following inequalities, which follows easily from the Taylor expansion of log(1 + x ) e a − a n ≤ (1 + an ) n ≤ e a for | a | ≤ n/ . We apply the above inequalities to estimate the integral, and obtain Z C x χ ( u ) χ ( v ) (cid:18) − r (˜ u + ˜ v ) + ˜ u ˜ vs − r (cid:19) n − dl = (1 + o (1)) Z C x e (cid:16) − nr (˜ u +˜ v ) s − r (cid:17) χ ( u ) χ ( v ) dl. We approximate the curve C x with the following line˜ u + ˜ v = 2 (cid:18)r s − x − x − r (cid:19) + o (cid:18) n (cid:19) . It follows that Z C x e (cid:16) − nr (˜ u +˜ v ) s − r (cid:17) χ ( u ) χ ( v ) dl = (1 + o (1))2 √ (cid:18) δ + r s − x − x − r (cid:19) + e − nr (cid:18) √ s − x − x − r (cid:19) s − r . Therefore, µ ( x ; s, χ ) = (1 + o (1)) 2 √ (cid:18) δ + r s − x − x − r (cid:19) + (cid:18)(cid:18) − x x (cid:19) (cid:0) s − r (cid:1) (cid:19) n − e − nr (cid:18) √ s − x − x − r (cid:19) s − r . This completes the proof of our proposition. (cid:3)
Conditional density for sphere packings.
In this section, we describe and estimate theprobability density function for sphere packings. Let s ′ = cos( θ ′ ) and r = √ − s ′ ) , where 1 / ≤ s ′ < /
2. Let 0 < δ = c n for some fixed c > χ ( y ) := ( ≤ y ≤ r + δ, . et x , y be two randomly independently chosen points on R n − with respect to the Euclideanmeasure such that | x | , | y | ≤ r + δ, where | . | is the Euclidean norm. Let α := arccos h x , y i ,U := | x | ,V := | y | , and T := | x − y | . The pushforward measure onto (
U, V, α ) has the following density µ ( U, V, α ) = U n − V n − sin( α ) n − dU dV dα, up to a positive scalar scalar that depends only on n . Similarly, the projection onto ( U, V, T ) hasthe following density function up to a positive scalar µ ( U, V, T ) = (
U V T )∆(
U, V, T ) n − dU dV dT, where ∆( U, V, T ) is the Euclidean area of the triangle with sides
U, V, T.
Let x := cos( α ) = U + V − T U V .
Let µ ( x ; χ ) be the induced density function on x subjected to the conditions T = 1 , and U, V ≤ r + δ .We define for x x + = ( x for x ≥ , r := p − (1 − x )( r + δ ) + x ( r + δ ) . Proposition 7.2.
Suppose that | x − s ′ | ≤ c n , x < / and δ = c n for some fixed < c < , < c < . . We have µ ( x ; χ ) = (1 + E )(1 − x ) n − (cid:18) − x ) (cid:19) n − vuut x − (1 − x ) r p − (1 − x ) r ! ( r + δ − r ) + up to a positive scalar multiple making this a probability measure on [ − , , and where | E | < (4 c +2 c +2) n for n ≥ .Proof. We have µ ( x ; χ ) = Z C x χ ( U ) χ ( V ) µ ( U, V, dl, where the integral is over the curve C x ⊂ R that is given by U + V − UV = x and dl is the inducedEuclidean metric on C x . We note that µ ( U, V,
1) =
U V (cid:0) (1 − x ) U V (cid:1) n − . Hence, µ ( x ; χ ) = (1 − x ) n − Z C x χ ( U ) χ ( V ) ( U V ) n − dl. Suppose that U and V are in the support of χ. Let U = r + ˜ U and V = r + ˜ V , then ˜
U , ˜ V ≤ c n . Wehave ˜ U + ˜ V = 12 r (1 − x ) − r + O (1 /n ) = ( x − s ′ )2 r (1 − x ) + O (1 /n ) . ince | x − s ′ | ≤ c n , and 0 . < r (1 − x ) , it follows that − c +2 c n < ˜ U , ˜ V ≤ c n for n ≥ U + ˜ V = 12 r (1 − x ) − r + E . where | E | = (cid:12)(cid:12)(cid:12)(cid:12) r (1 − x ) (cid:16) x ˜ U ˜ V − ˜ U − ˜ V (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c + 2 c ) n . We have
U V = r + r ( ˜ U + ˜ V ) + ˜ U ˜ V = r U + ˜ Vr ! + ˜ U ˜ V = 12(1 − x ) + E , where | E | < ( c +2 c ) n . Hence, µ ( x ; χ ) = (1 − x ) n − (cid:18) − x ) (cid:19) n − Z C x (cid:18) E − x ) (cid:19) n − χ ( U ) χ ( V ) dl. Note that (cid:18) E − x ) (cid:19) n − = 1 + E ′ , where E ′ ≤ ( c +2 c ) n for n ≥ Z C x (cid:18) E − x ) (cid:19) n − χ ( U ) χ ( V ) dl = (1 + ¯ E ) Z C x χ ( U ) χ ( V ) dl, where ¯ E ≤ ( c +2 c ) n . We parametrize the curve C x with V to obtain U ( V ) = p − (1 − x ) V + xV. We have dUdV = x − (1 − x ) V p − (1 − x ) V = x − (1 − x ) r p − (1 − x ) r + E , where | E | = (cid:12)(cid:12)(cid:12)(cid:12) ( V − r ) (1 − x )(1 − (1 − x ) V ) / (cid:12)(cid:12)(cid:12)(cid:12) for some V ∈ ( r − c +2 c n , r + c n ) , which implies (cid:12)(cid:12)(cid:12) (1 − x )(1 − (1 − x ) V ) / (cid:12)(cid:12)(cid:12) < . Hence, | E | ≤ (cid:18) c + 2 c n (cid:19) . Hence, Z C x χ ( U ) χ ( V ) dl = + Z r + δr s (cid:18) dUdV (cid:19) dV = vuut x − (1 − x ) r p − (1 − x ) r ! ( r + δ − r ) + (1 + ¯ E )where r := p − (1 − x )( r + δ ) + x ( r + δ ) , and | E | ≤ (cid:18) c + 2 c n (cid:19) . he first equality = + above means equal to 0 if r + δ < r . Therefore, µ ( x ; χ ) = (1 + E )(1 − x ) n − (cid:18) − x ) (cid:19) n − vuut x − (1 − x ) r p − (1 − x ) r ! ( r + δ − r ) + up to a positive scalar multiple, where | E | < (4 c +2 c +2) n for n ≥ (cid:3) The critical function
In this section, we give a proof of Theorems 1.7 and 1.10. Consider the function g ( t ; s, λ ) := ( t − s ) η ( t ) d − X i =0 λ i ( t − s ) det (cid:20) ˜ p i +1 ( t ) ˜ p i +1 ( s )˜ p i ( t ) ˜ p i ( s ) (cid:21)! . We write g ( t ; s, λ ) = P d + h − i =0 g i ( s, λ ) p i ( t ).8.1. Computing g ( s, λ ) . We show that g ( s, λ ) as a quadratic form in λ has a diagonal form. Proposition 8.1.
We have g ( s, λ ) = − d − X j =0 ˜ a j +1 ˜ p j ( s )˜ p j +1 ( s ) λ j . As a result for every ≤ i ≤ m − , we have ∂g ∂λ i = − a i +1 λ i ˜ p i ( s )˜ p i +1 ( s ) . Proof.
We have g ( s, λ ) = Z − g ( t ; s, λ ) dµ α ( t )= Z − d − X j =0 d − X l = j ˜ a l +1 λ l ˜ p j ( t )˜ p j ( s ) × λ d − ˜ p d − ( s )˜ p d ( t ) + d − X i =0 ( − λ i ˜ p i +1 ( s ) + λ i − ˜ p i − ( s ))˜ p i ( t ) ! η ( t ) dµ α ( t )= d − X j =0 d − X l = j ˜ a l +1 λ l ˜ p j ( s )( − λ j ˜ p j +1 ( s ) + λ j − ˜ p j − ( s ))= d − X j =0 λ j ˜ p j ( s )˜ p j +1 ( s ) d − X l = j +1 ˜ a l +1 λ l − m − X l = j ˜ a l +1 λ l = − d − X j =0 ˜ a j +1 ˜ p j ( s )˜ p j +1 ( s ) λ j . Hence, we also have ∂g ∂λ i = − a i +1 λ i ˜ p i ( s )˜ p i +1 ( s ) . This completes the proof of our proposition. (cid:3) .2. Lagrange multiplier.
Let R ( s, λ ) := g (1; s, λ ) g ( s, λ ) . Note that for fixed s , R ( s, λ ) is invariant under multiplying λ with a scalar. So, for fixed s , wemay consider R ( s, λ ) as a function on the projective space P d − ( R ) . We define(40) λ c := ( λ c , . . . , λ cd − ) , where λ ci = 1˜ a i +1 (cid:18) ˜ p i (1)˜ p i ( s ) − ˜ p i +1 (1)˜ p i +1 ( s ) (cid:19) . We prove Theorem 1.7 in the following.
Proof of Theorem 1.7.
We apply a version of the Lagrange multiplier method and show that λ c (upto scalar) is the unique critical point of R ( s, λ ) subjected to g > . Since R ( s, λ ) is a function onthe projective space, without loss of generality we may assume g ( s, λ ) = 1 . So, minimizing R ( s, λ )subjected to g > g (1; s, λ ) = 11 − s d − X i =0 λ i det (cid:20) ˜ p i +1 (1) ˜ p i +1 ( s )˜ p i (1) ˜ p i ( s ) (cid:21)! on the quadric g ( s, λ ) = 1 . First, we show that ( λ c , . . . , λ cd − ) is a critical point for the restrictionof R on g ( s, λ ) = 1 . We have ∇ R = 1 g ∇ g (1; s, λ ) − g (1; s, λ ) g ∇ g . Therefore, λ is a critical point for the restriction of R on g ( s, λ ) = 1 if and only if ∇ g (1; s, λ )is parallel to ∇ g . In other words, it is enough to show that ∇ g (1; s, λ ) = ∇ g as points in theprojective space P d − ( R ) . In what follows we consider vectors as elements of P d − ( R ) , so we ignorethe scalars. ∇ g (1; s, λ ) = (cid:18) ∂g (1; . ) ∂λ i (cid:19) ≤ i ≤ d − = (cid:18)P d − i =0 λ i det (cid:20) ˜ p i +1 (1) ˜ p i +1 ( s )˜ p i (1) ˜ p i ( s ) (cid:21)(cid:19) − s (cid:18) det (cid:20) ˜ p i +1 (1) ˜ p i +1 ( s )˜ p i (1) ˜ p i ( s ) (cid:21)(cid:19) ≤ i ≤ d − = (cid:18) det (cid:20) ˜ p i +1 (1) ˜ p i +1 ( s )˜ p i (1) ˜ p i ( s ) (cid:21)(cid:19) ≤ i ≤ d − ∈ P d − ( R ) . By Proposition 8.1, we have ∇ g ( s, λ ) = (˜ a i +1 λ i ˜ p i ( s )˜ p i +1 ( s )) ≤ i ≤ d − ∈ P d − ( R ) . If λ = ( λ i ) ≤ i ≤ d − is a critical point then( λ i ) ≤ i ≤ d − = (cid:18) a i +1 (˜ p i (1) / ˜ p i ( s ) − ˜ p i +1 (1) / ˜ p i +1 ( s )) (cid:19) ≤ i ≤ d − = λ c ∈ P d − ( R ) . This implies that λ c ∈ P d − ( R ) is the unique critical point for R subjected to g > . If we additionally have d ( s ) ≤ d < d ( s ), then the quadratic form g ( s, λ ) has signature (1 , d − . Therefore the set C := { λ : g ( s, λ ) ≥ } s a convex set. Furthermore, note that d − X i =0 λ ci det (cid:20) ˜ p i +1 (1) ˜ p i +1 ( s )˜ p i (1) ˜ p i ( s ) (cid:21) = − d − X i =0 ˜ a i +1 ˜ p i ( s )˜ p i +1 ( s )( λ ci ) = g [ µ,d,η ]0 > , where the final equality follows from Proposition 8.1. Therefore, the tangent hyperplane of thequadric g ( s, λ ) = 1 at λ c separates the origin and the quadric. Hence, it follows that λ c is theunique global minimum of R. This concludes the proof of Theorem 1.7. (cid:3)
We now prove Theorem 1.10 providing us with a clean expression for f [ µ,d,η ] . Proof of Theorem 1.10.
Recall that f [ µ,d,η ] ( t ) := d − X i =0 λ c i t − s det (cid:20) ˜ p i +1 ( t ) ˜ p i +1 ( s )˜ p i ( t ) ˜ p i ( s ) (cid:21) . We have for every i ≤ d − Z − f [ µ,d,η ] ( t )˜ p i ( t )( t − s ) η ( t ) dµ = − λ c i ˜ p i +1 ( s ) + λ c i − ˜ p i − ( s )= 1˜ a i +1 (cid:18) ˜ p i +1 (1) − ˜ p i (1)˜ p i +1 ( s )˜ p i ( s ) (cid:19) + 1˜ a i (cid:18) ˜ p i − (1) − ˜ p i (1)˜ p i − ( s )˜ p i ( s ) (cid:19) = 1˜ p i ( s ) (cid:18) a i +1 det (cid:20) ˜ p i +1 (1) ˜ p i +1 ( s )˜ p i (1) ˜ p i ( s ) (cid:21) − a i det (cid:20) ˜ p i (1) ˜ p i ( s )˜ p i − (1) ˜ p i − ( s ) (cid:21)(cid:19) = (1 − s )˜ p i ( s ) i X j =0 ˜ p j (1)˜ p j ( s ) − i − X j =0 ˜ p j (1)˜ p j ( s ) = (1 − s )˜ p i (1) . This implies that(41) Z − f [ µ,d,η ] ( t ) q ( t )( t − t − s ) η ( t ) dµ = 0 , where q ( t ) is any polynomial of degree at most d − . Note that f [ µ,d,η ] ( t ) is uniquely determined(up to a constant) by being a polynomial of degree d − t − t − s ) det ˜ p d +1 ( t ) ˜ p d +1 ( s ) ˜ p d +1 (1)˜ p d ( t ) ˜ p d ( s ) ˜ p d (1)˜ p d − ( t ) ˜ p d − ( s ) ˜ p d − (1) , is of degree d − f [ µ,d,η ] ( t ) = c ( t − t − s ) det ˜ p d +1 ( t ) ˜ p d +1 ( s ) ˜ p d +1 (1)˜ p d ( t ) ˜ p d ( s ) ˜ p d (1)˜ p d − ( t ) ˜ p d − ( s ) ˜ p d − (1) , where c ∈ C . Similarly, we show that d − X i =0 ˜ p i ( t ) det (cid:20) ˜ p d ( s ) ˜ p d (1)˜ p i ( s ) ˜ p i (1) (cid:21) atisfies equation (41). It is enough to check it for every q l ( t ) = 1( t − t − s ) det ˜ p l +1 ( t ) ˜ p l +1 ( s ) ˜ p l +1 (1)˜ p l ( t ) ˜ p l ( s ) ˜ p l (1)˜ p l − ( t ) ˜ p l − ( s ) ˜ p l − (1) , where l < d. We have Z − d − X i =0 ˜ p i ( t ) det (cid:20) ˜ p d ( s ) ˜ p d (1)˜ p i ( s ) ˜ p i (1) (cid:21)! det ˜ p l +1 ( t ) ˜ p l +1 ( s ) ˜ p l +1 (1)˜ p l ( t ) ˜ p l ( s ) ˜ p l (1)˜ p l − ( t ) ˜ p l − ( s ) ˜ p l − (1) η ( t ) dµ = det det (cid:20) ˜ p d ( s ) ˜ p d (1)˜ p l +1 ( s ) ˜ p l +1 (1) (cid:21) ˜ p l +1 ( s ) ˜ p l +1 (1)det (cid:20) ˜ p d ( s ) ˜ p d (1)˜ p l ( s ) ˜ p l (1) (cid:21) ˜ p l ( s ) ˜ p l (1)det (cid:20) ˜ p d ( s ) ˜ p d (1)˜ p l − ( s ) ˜ p l − (1) (cid:21) ˜ p l − ( s ) ˜ p l − (1) = 0 , where that last identity follows from the fact that the first column is a linear combination of theother two columns. This concludes the proof of our theorem. (cid:3) Proof of Theroem 1.12.
Our method is based on the proof of Theroem 1.10. It is enough to showthat b [ µ,d,η ] ( t ) and r [ µ,d,η ] ( t ) are of degree d − d − b [ µ,d,η ] ( t ) satisfy equation (41). Similarly, it is enough to show that r [ µ,d,η ] ( t )satisfies equation (41) for every q i ( t ) := 1( t − α ) . . . ( t − α h ) det p i + k ( t ) p i + k ( α ) . . . p i + k ( α h ) p i + k − ( t ) p i + k − ( α ) . . . p i + k − ( α h )... ... ... p i ( t ) p i ( α ) . . . p i ( α h ) , where i < d − . This follows from a similar argument as in the proof of Theorem 1.10. (cid:3)
We end this section with a computation of the γ i.j . Lemma 8.2. γ i,j = ( p i (1) p j (1) P i − l = j a l +1 p l (1) p l +1 (1) for j < i, otherwise.Proof. We will prove this using the Christoffel-Darboux formula (16). Indeed, we may rewrite γ i,j = p i (1) Z − p i ( t ) p i (1) − t − p j ( t ) dµ = p i (1) i − X l =0 Z − p l +1 ( t ) p l +1 (1) − p l ( t ) p l (1) t − p j ( t ) dµ = p i (1) i − X l = j Z − p l +1 ( t ) p l +1 (1) − p l ( t ) p l (1) t − p j ( t ) dµ. Using the Christoffel-Darboux formula (16), we may write p l +1 ( t ) p l +1 (1) − p l ( t ) p l (1) t − a l +1 p l (1) p l +1 (1) l X k =0 p k (1) p k ( t ) . ubstituting this into the above, we obtain γ i,j = p i (1) i − X l = j l X k =0 Z − a l +1 p k (1) p l (1) p l +1 (1) p k ( t ) p j ( t ) dµ = p i (1) p j (1) i − X l = j a l +1 p l (1) p l +1 (1) . (cid:3) Critical value of functional
In this section, we give a proof of Theorem 1.14. We now take our function g [ µ,d,η ] for generalpolynomials η . By Theorem 1.12, up to a nonzero constant , f [ µ,d,η ] has the equivalent expressions b [ µ,d,η ] ( t ) = (cid:16)Q d + h +1 i = d a i (cid:17) r [ µ,d,η ] , where b [ µ,d,η ] and r [ µ,d,η ] are defined in equations (12) and (13),respectively. Consider the function h [ µ,d,η ] ( t ) := ( t − s ) η ( t ) b [ µ,d,η ] ( t ) .h [ µ,d,η ] is, up to a constant multiple, equal to g [ µ,d,η ] , and so L ( g [ µ,d,η ] ) = L ( h [ µ,d,η ] ). We nowcompute the value of the functional L at h [ µ,d,η ] . First, let us compute h [ µ,d,η ]0 . Lemma 9.1. h [ µ,d,η ]0 = b [ µ,d,η ] (1) det p d + h +1 (1) P d + hl = d − a l +1 p l (1) p l +1 (1) p d + h +1 ( s ) p d + h +1 (1) p d + h +1 ( α ) . . . p d + h +1 ( α h ) ... ... p d (1) P d − l = d − a l +1 p l (1) p l +1 (1) p d ( s ) p d (1) p d ( α ) . . . p d ( α h )0 p d − ( s ) p d − (1) p d − ( α ) . . . p d − ( α h ) . Proof.
By definition, h [ µ,d,η ]0 = Z − h [ µ,d,η ] ( t ) dµ ( t ) . Let c ′ := Q d + h +1 i = d a i . Using the explicit expression for h [ µ,d,η ] , we obtain h [ µ,d,η ]0 = c ′ Z − ( t − s ) η ( t ) b [ µ,d,η ] ( t ) r [ µ,d,η ] ( t ) dµ = c ′ Z − t − p d + h +1 ( t ) p d + h +1 ( s ) p d + h +1 (1) p d + h +1 ( α ) . . . p d + h +1 ( α h )... p d ( t ) p d ( s ) p d (1) p d ( α ) . . . p d ( α h ) p d − ( t ) p d − ( s ) p d − (1) p d − ( α ) . . . p d − ( α h ) r [ µ,d,η ] ( t ) dµ ( t )= c ′ Z − d + h X j =0 det γ d + h +1 ,j p d + h +1 ( s ) p d + h +1 (1) p d + h +1 ( α ) . . . p d + h +1 ( α h )... ... γ d,j p d ( s ) p d (1) p d ( α ) . . . p d ( α h ) γ d − ,j p d − ( s ) p d − (1) p d − ( α ) . . . p d − ( α h ) p j ( t ) r [ µ,d,η ] ( t ) dµ ( t ) . ote that Lemma 8.2 gives us that for j < d ,det γ d + h +1 ,j p d + h +1 ( s ) p d + h +1 (1) p d + h +1 ( α ) . . . p d + h +1 ( α h )... ... γ d,j p d ( s ) p d (1) p d ( α ) . . . p d ( α h ) γ d − ,j p d − ( s ) p d − (1) p d − ( α ) . . . p d − ( α h ) = p j (1) det p d + h +1 (1) P d + hl = j a l +1 p l (1) p l +1 (1) p d + h +1 ( s ) p d + h +1 (1) p d + h +1 ( α ) . . . p d + h +1 ( α h )... ... p d (1) P d − l = j a l +1 p l (1) p l +1 (1) p d ( s ) p d (1) p d ( α ) . . . p d ( α h ) p d − (1) P d − l = j a l +1 p l (1) p l +1 (1) p d − ( s ) p d − (1) p d − ( α ) . . . p d − ( α h ) = p j (1) det p d + h +1 (1) P d + hl = d − a l +1 p l (1) p l +1 (1) p d + h +1 ( s ) p d + h +1 (1) p d + h +1 ( α ) . . . p d + h +1 ( α h )... ... p d (1) P d − l = d − a l +1 p l (1) p l +1 (1) p d ( s ) p d (1) p d ( α ) . . . p d ( α h )0 p d − ( s ) p d − (1) p d − ( α ) . . . p d − ( α h ) , where the last equality follows from subtracting a multiple of the third colum from the first column.Denote by A the determinant expression in the final line. A is independent of j . Using this andthe computation above for h [ µ,d,η ]0 , we obtain h [ µ,d,η ]0 = c ′ A Z − d + h X j =0 p j (1) p j ( t ) r [ µ,d,η ] ( t ) dµ = c ′ Ar [ µ,d,η ] (1) = Ab [ µ,d,η ] (1) . The conclusion follows. (cid:3)
Combining the previous results, we obtain Theorem 1.14.10.
Positivity of Fourier coefficients
Proof of Theorem 1.13.
By definition 1.9 and Theorem 1.12, the inequalities (11) and (15) areequivalent. Next, we show that the inequalities (10) and the inequalities (14) are equivalent. For j > d + h, p j is orthogonal to any polynomial of degree at most d + h , hence Z − p j ( t )( t − s ) η ( t ) cf [ µ,d,η ] ( t ) dµ = 0 . This implies inequalities (10) for j > d + h . Suppose that d − ≤ j ≤ d + h . By the definition of γ i,j . det γ d + h +1 ,j p d + h +1 ( s ) p d + h +1 (1) p d + h +1 ( α ) . . . p d + h +1 ( α h )... ... γ d,j p d ( s ) p d (1) p d ( α ) . . . p d ( α h ) γ d − ,j p d − ( s ) p d − (1) p d − ( α ) . . . p d − ( α h ) and Z − t −
1) det p d + h +1 ( t ) p d + h +1 ( s ) p d + h +1 (1) p d + h +1 ( α ) . . . p d + h +1 ( α h )... p d ( t ) p d ( s ) p d (1) p d ( α ) . . . p d ( α h ) p d − ( t ) p d − ( s ) p d − (1) p d − ( α ) . . . p d − ( α h ) p j ( t ) dµ re equal. This implies that the inequalities (10) and the inequalities (14) are equivalent for d − ≤ j ≤ d + h . Note that for j < i , we have γ i,j = Z − t − (cid:20) p j (1) p i (1) p j ( t ) p i ( t ) (cid:21) dµ Hence, for j < d det γ d + h +1 ,j p d + h +1 ( s ) p d + h +1 (1) p d + h +1 ( α ) . . . p d + h +1 ( α h )... ... γ d,j p d ( s ) p d (1) p d ( α ) . . . p d ( α h ) γ d − ,j p d − ( s ) p d − (1) p d − ( α ) . . . p d − ( α h ) = p j (1) D where,(42) D := Z − t −
1) det p d + h +1 ( t ) p d + h +1 ( s ) p d + h +1 (1) p d + h +1 ( α ) . . . p d + h +1 ( α h )... p d ( t ) p d ( s ) p d (1) p d ( α ) . . . p d ( α h ) p d − ( t ) p d − ( s ) p d − (1) p d − ( α ) . . . p d − ( α h ) dµ. Inequality (10) for j = d − p d − (1), implies that κD > . Therefore, theinequalities (10) for j < d − j = d − . This completes the proof of our theorem. (cid:3)
Corollary 10.1.
If the measure µ satisfies the Krein condition and s is such that p d ( s ) < and p d ( s ) p d (1) − p i ( s ) p i (1) ≤ for ≤ i ≤ d , then η ( t ) := 1 is ( µ, s, d ) -positive.Proof. Since p d ( s ) p d (1) − p i ( s ) p i (1) ≤ ≤ i ≤ d, det (cid:20) p d ( s ) p d (1) p i ( s ) p i (1) (cid:21) ≤ ≤ i ≤ d − j = d det γ d +1 ,j p d +1 ( s ) p d +1 (1) γ d,j p d ( s ) p d (1) γ d − ,j p d − ( s ) p d − (1) ≤ . Since p d ( s ) <
0, it is easy to check that
D < γ d +1 ,j p d +1 ( s ) p d +1 (1) γ d,j p d ( s ) p d (1) γ d − ,j p d − ( s ) p d − (1) ≤ . for every 0 ≤ j ≤ d − . By Theorem 1.13, η ( t ) := 1 is ( µ, s, d )-positive. (cid:3) Remark . In particular, dµ α,β satisfies the Krein condition whenever α ≥ β , and so Corollary 10.1is true for the corresponding Jacobi polynomials. This recovers the extremal polynomials of Lev-enshtein of odd degree. Corollary 10.2.
For η = 1 + t and µ = dµ α , if d is such that ˜ p d ( s ) < and ˜ p j ( s )˜ p j (1) − ˜ p j +1 ( s )˜ p j +1 (1) ≥ forevery ≤ j ≤ d − , then η is ( dµ α , s, d ) -positive.Proof. By Theorem 1.13 it is enoguh to verify inequalities (14) and (15) for some κ ∈ C . First, wecheck inequality (15). The inequality (15) is equivalent to Z − p i ( t )( t − s )(1 + t ) f [ µ,d,η ] ( t ) dµ α ≥ . Let ˜ p i be L -normalized versions of the Jacobi polynomials p α,α +1 i which satisfy(44) p α,α +1 n ( t ) = ( n + 1)( n + α + 1)(1 + t ) p α,αn +1 ( t ) + p α,αn ( t )1 + t . ote that the coefficients on the right hand side of the above are positive. Hence, it is enough toshow that ( t − s ) f [ dµ α ,d, t ] ( t ) has non-negative coefficients when expanded in the basis ˜ p i . As inthe proof of Corollary 10.1, this is true since ˜ p d ( s ) < ˜ p d ( s )˜ p d (1) − ˜ p d − ( s )˜ p d − (1) ≤ p d +1 ( s ) p d +1 (1) p d +1 ( − p d ( s ) p d (1) p d ( − p j ( s ) p j (1) p j ( − have the same sign as j varies. Since, p j ( − t ) = ( − j p j ( t ) , we are reduced to showing that thedeterminants det p j ( s ) p j (1) p d ( s ) p d (1) p d +1 ( s ) p d +1 (1) ( − j ( − d ( − d +1 have the same sign as j varies. We split this verification into two cases based on the parity of d . If d is even, this determinant is det p j ( s ) p j (1) p d ( s ) p d (1) p d +1 ( s ) p d +1 (1) ( − j − . For j even, this reduces todet p j ( s ) p j (1) p d ( s ) p d (1) p d +1 ( s ) p d +1 (1) − = det p j ( s ) p j (1) p d ( s ) p d (1) p d +1 ( s ) p d +1 (1) − = 2 (cid:18) p j ( s ) p j (1) − p d ( s ) p d (1) (cid:19) . For odd j (and d even), we havedet p j ( s ) p j (1) p d ( s ) p d (1) p d +1 ( s ) p d +1 (1) − − = det p j ( s ) p j (1) p d ( s ) p d (1) p d +1 ( s ) p d +1 (1) − − = 2 (cid:18) p j ( s ) p j (1) − p d +1 ( s ) p d +1 (1) (cid:19) . If d is chosen so that these two quantities are non-negative, then we have the ( dµ α , s, d )-positivityof 1 + t . In the case of Levenshtein, d is the chosen to be the first index at which ˜ p j ( s )˜ p j (1) − ˜ p j +1 ( s )˜ p j +1 (1) changes sign from positive to negative. This quantity is, after multiplication by 1 + s , equal to p j ( s ) p j (1) − p j +2 ( s ) p j +2 (1) . The equality follows from relation (44). This implies that for d even, the determinants above arenon-negative for every 0 ≤ j ≤ d −
1. Similarly, we can settle the case when d is odd. (cid:3) Therefore, the positivity conditions we impose subsume those of Levenshtein.11.
Numerics
In this section, we give a list of improvement factors to maximal sphere packing densities whenusing the linear programming method. The table consists of our factors of improvement to theright hand side of the inequality (4) δ n ≤ sin n ( θ/ M ( n, θ ) f Cohn and Zhao as θ varies between 61 ◦ and 90 ◦ and when when M ( n, θ ) is bounded usingLevenshtein’s optimal polynomials [Lev79a]. For each given n and θ , the factor of improvement is,by Proposition 4.7, (cid:18) δr ( θ ) (cid:19) − n , where δ > H constructed in this paper in Section 4 with r = r ( θ ) := √ − cos θ ) , F = χ [0 ,r ( θ )+ δ ] , and such that r + δ ≤
1. In Theorem 1.6, we restricted to the case wherecos θ ′ were roots of Jacobi polynomials for the simplicity of computations in Section 5. When doingnumerics, however, in order to find the maximal δ , no such restriction is necessary; the followingtable neither needs nor imposes such a restriction. As all quantities but δ are easily computable orgiven, Table 2 may be used to compute the value of the appropriate δ . Though we do not includedimensions higher than 130, our preliminary numerics suggest that our 0 . ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Table 2.
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Institute for Advanced Study, Princeton, NJ, USA
Email address : [email protected] Department of Mathematics, University of Regensburg, Regensburg, Germany
Email address : [email protected]@ur.de