Spherical two-distance sets

Spherical t w o-distance sets Oleg R. Musin ∗ Abstract A set S of unit v ectors in n -dimensional Euclidean space is called spherical tw o-distance set, if there a re tw o num b ers a and b so that the inner products of distinct vectors of S are either a or b . It is known that the largest cardinali ty g ( n ) o f spherical tw o-distance sets does not exceed n ( n + 3) / 2 . This up per b ound is known to b e tight for n = 2 , 6 , 22. The set of mid-p oints of the edges of a regular simplex gives the lo wer b ound L ( n ) = n ( n + 1) / 2 for g ( n ). In th is pap er u sing the so-called p olynomial method it is prov ed that for nonnegative a + b the largest cardin alit y of S is not greater than L ( n ). F o r th e case a + b < 0 we prop ose upp er b ounds on | S | which are based on Delsarte’s method . Using this we show that g ( n ) = L ( n ) for 6 < n < 22 , 23 < n < 40, and g (23) = 276 or 277. 1 In tro duction A set S in Euclidean space R n is called a two-distanc e set , if there ar e tw o distances c and d , and the distances b et ween pairs of points of S a re either c or d . If a t wo-distance set S lies in the unit s phere S n − 1 , then S is called spheric al two-distanc e set . In other words, S is a set of unit vectors, there ar e tw o real nu mbers a and b , − 1 ≤ a, b < 1, and inner pro ducts of distinct vectors of S are either a or b . The r atios o f distances o f tw o-distanc e sets are quite res trictiv e. Namely , Larman, Rogers, and Seidel [8 ] hav e prov ed the following fact: if the ca rdinality of a tw o- distance se t S in R n , with distanc e s c and d, c < d, is gre ater than 2 n + 3, then the ratio c 2 /d 2 equals ( k − 1) /k for an in teger k with 2 ≤ k ≤ 1 + √ 2 n 2 . Einhorn a nd Schoenberg [6] pro ved that there are finitely man y t wo-distance sets S in R n with cardinality | S | ≥ n + 2. Delsarte, Go ethals , and Seidel [5] prov ed that the largest ca rdinality of spherical tw o-distance sets in R n (w e denote it by g ( n )) is b ounded by n ( n + 3) / 2, i.e., g ( n ) ≤ n ( n + 3) 2 . ∗ Departmen t of Mathemat ics, Univ ersi t y of T exas at Bro wnsvill e. oleg.musin@utb.edu 1 Moreov er , they giv e examples of s pher ical t wo-distance sets with n ( n + 3) / 2 po in ts for n = 2 , 6 , 2 2. (Therefor e, in these dimensions we ha ve equality g ( n ) = n ( n + 3) / 2 .) Blo ckh uis [2] s howed that the cardinality o f (Euclidean) t wo- distance sets in R n do es not exceed ( n + 1)( n + 2) / 2. The standar d unit vectors e 1 , . . . , e n +1 form an orthog onal basis of R n +1 . Denote by ∆ n the reg ular simplex with vertices 2 e 1 , . . . , 2 e n +1 . Let Λ n be the set of points e i + e j , 1 ≤ i < j ≤ n + 1 . Since Λ n lies in the h yp erpla ne P x k = 2, we see that Λ n represents a spherical tw o- distance set in R n . O n the other hand, Λ n is the set of mid-p oints of the edges of ∆ n . Thus, g ( n ) ≥ | Λ n | = n ( n + 1) 2 . F or n < 7 the largest cardinality of Euclidean tw o-dista nc e sets is g ( n ), where g (2) = 5 , g (3) = 6 , g (4) = 10 , g (5) = 16 , and g (6 ) = 27 (s e e [1 0]). Ho wever, for n = 7 , 8 Lison ˇ ek [10] discovered no n-spherical maxima l tw o-distance s ets of the cardinality 29 and 45 r e spectively . In this pap er we prove that g ( n ) = n ( n + 1) 2 , where 6 < n < 40 , n 6 = 22 , 23 , and g (23) = 276 o r 2 77. This pro o f (Section 4) is based on the new sharp upper bo und  n +1 2  for spheric a l tw o-distance sets with a + b ≥ 0 (Section 2), and on the Delsarte b ounds for spherica l tw o-distance sets in the case a + b < 0. 2 Linearly indep enden t p olynomials The upp er b ound n ( n + 3) / 2 for spherical tw o- distance sets [5], the b ound  n +2 2  for Euclidea n tw o-distance sets [2], as well a s the b ound  n + s s  for s − dis tance sets [1, 3] were obtained by the polynomia l metho d. The main idea of this metho d is the following: to ass oc iate sets to p olynomials a nd show that thes e po lynomials are linearly independent as members o f the co rresp onding vector space. Now we apply this idea to improve upper b ounds for spherical t wo-distance sets with a + b ≥ 0 . Theorem 1. L et S b e a spheric al two-distanc e set in R n with inner pr o du ct s a and b . If a + b ≥ 0 , t hen | S | ≤ n ( n + 1) 2 . Pr o of. Let F ( t ) := ( t − a )( t − b ) (1 − a )(1 − b ) . F or a unit vector y ∈ R n we define the function F y : S n − 1 → R by F y ( x ) := F ( h x, y i ) , x ∈ R n , || x || = 1 . 2 Let S = { x 1 , . . . , x m } be an m -element set. Denote f i ( x ) := F x i ( x ). Since f i ( x j ) = δ i,j , (1) the quadratic p olynomials f i , i = 1 , . . . , m, ar e line a rly indep endent. Let e 1 , . . . , e n be a ba sis of R n . Let L i ( x ) := h x, e i i , x ∈ S n − 1 . Then the linear p olynomials L 1 , . . . , L n are also linearly indep endent. Now we show that if a + b ≥ 0, then f 1 , . . . , f m , L 1 , . . . , L n form a linear ly independent system of p olynomials. Assume the c o n verse. Then n X k =1 d k L k ( x ) = m X i =1 c i f i ( x ) , where there are nonzero d k and c i . Let v = d 1 e 1 + . . . + d n e n . Then h x, v i = X i c i f i ( x ) . (2) F or x = x i in (2 ), using (1), we g e t c i = h x i , v i . T ake x = v and x = − v in (2). Then we have || v || 2 = X i c i f i ( v ) = X i c i F ( c i ) , (3) −|| v || 2 = X i c i f i ( − v ) = X i c i F ( − c i ) . (4) Subtracting (3) from (4), we o btain −|| v || 2 = a + b (1 − a )(1 − b ) X i c 2 i . This yields v = 0, a contradiction. Note that the dimensio n of the vector spac e of quadratic polyno mials o n the sphere S n − 1 is n ( n + 3) / 2. Therefore, dim { f 1 , . . . , f m , L 1 , . . . , L n } = m + n ≤ n ( n + 3) 2 . Thu s, | S | = m ≤ n ( n + 1) / 2 . Denote b y ρ ( n ) the larg est p ossible cardinality of spherical tw o-distanc e sets in R n with a + b ≥ 0 . 3 Theorem 2. If n ≥ 7 , then ρ ( n ) = n ( n + 1) 2 . Pr o of. Theorem 1 implies that ρ ( n ) ≤ n ( n + 1) / 2. On the other hand, the set of mid-po in ts of the edges of a reg ular simplex has n ( n + 1) / 2 points a nd a + b ≥ 0 for n ≥ 7. Indeed, for this spher ical tw o-dista nc e set we have a = n − 3 2( n − 1) , b = − 2 n − 1 . Thu s, a + b = n − 7 2( n − 1) ≥ 0 . 3 Delsarte’s metho d for t w o-distance sets Delsarte’s method is widely used in co ding theory and discrete geometry for finding b ounds for error -correcting co de s , spherical co des, and spher e packings (see [4, 5, 7]). In this metho d upp er b ounds for spherical co des ar e giv en by the following theorem: Theorem 3 ([5, 7]) . L et T b e a subset of the interval [ − 1 , 1] . L et S b e a set of unit ve ct ors in R n such that the set of inner pr o ducts of distinct ve ctors of S lies in T . Supp ose a p olynomial f is a n onne gative line ar c ombination of Ge genb auer p olynomials G ( n ) k ( t ) , i.e., f ( t ) = X k f k G ( n ) k ( t ) , wher e f k ≥ 0 . If f ( t ) ≤ 0 for al l t ∈ T and f 0 > 0 , then | S | ≤  f (1 ) f 0  There are many wa ys to define Gegenbauer (or ultraspherical) p olynomia ls G ( n ) k ( t ). G ( n ) k are a sp ecial case of Ja cobi p olynomials P ( α,β ) k with α = β = ( n − 3) / 2 and with normaliz a tion G ( n ) k (1) = 1. Another w ay to define G ( n ) k is the recurr e nce formula: G ( n ) 0 = 1 , G ( n ) 1 = t , . . . , G ( n ) k = (2 k + n − 4 ) t G ( n ) k − 1 − ( k − 1) G ( n ) k − 2 k + n − 3 . F or instance, G ( n ) 2 ( t ) = nt 2 − 1 n − 1 , 4 G ( n ) 3 ( t ) = ( n + 2) t 3 − 3 t n − 1 , G ( n ) 4 ( t ) = ( n + 2)( n + 4) t 4 − 6( n + 2) t 2 + 3 n 2 − 1 . Now for given n, a, b we intro duce poly no mials P i ( t ) , i = 1 , . . . , 5. i = 1 : P 1 ( t ) = ( t − a )( t − b ) = f (1) 0 + f (1) 1 t + f (1) 2 G ( n ) 2 ( t ) . i = 2 : P 2 ( t ) = ( t − a )( t − b )( t + c ) = f (2) 0 + f (2) 1 t + f (2) 2 G ( n ) 2 ( t ) + f (2) 3 G ( n ) 3 ( t ), where c is defined by the equa tion f (2) 1 = 0 . i = 3 : P 3 ( t ) = ( t − a )( t − b )( t + a + b ) = f (3) 0 + f (3) 1 t + f (3) 2 G ( n ) 2 ( t ) + f (3) 3 G ( n ) 3 ( t ) . Note that f (3) 2 = 0 . i = 4 : P 4 ( t ) = ( t − a )( t − b )( t 2 + c t + d ) = P f (4) k G ( n ) k ( t ), where c and d are defined by the equations f (4) 1 = f (4) 2 = 0 . i = 5 : P 5 ( t ) = ( t − a )( t − b )( t 2 + c t + d ) = P f (5) k G ( n ) k ( t ), where c and d are defined by the equations f (5) 2 = f (5) 3 = 0 . Denote by D ( n ) i the set of all pair s ( a, b ) such that the p olynomial P i ( t ) is well defined, all f ( i ) k ≥ 0, and f ( i ) 0 > 0 . F or insta nce, D ( n ) 1 =  ( a, b ) ∈ I 2 : f (1) 1 = − a − b ≥ 0 , f (1) 0 = ab + 1 n > 0  , D ( n ) 2 =  ( a, b ) ∈ I 2 : a + b 6 = 0 , c ≥ a + b, f (2) 0 = a bc + c − a − b n > 0  , where I = [ − 1 , 1) , c = ab ( n + 2) + 3 ( n + 2)( a + b ) . Let U ( n ) i ( a, b ) := P i (1) f ( i ) 0 . Note that we have P i ( a ) = P i ( b ) = 0. Then Theor em 3 yie lds Theorem 4. L et S b e a spheric al two-distanc e set in R n with inner pr o du ct s a and b . Supp ose ( a, b ) ∈ D ( n ) i for some i, 1 ≤ i ≤ 5 . Then | S | ≤ U ( n ) i ( a, b ) . Let S be a spherical tw o- distance set in R n with inner pro ducts a and b , where a > b . Let c = √ 2 − 2 a, d = √ 2 − 2 b . Then c and d are the Euclidean distances of S . Let b k ( a ) = k a − 1 k − 1 . 5 0 0.05 0.1 0.15 0.2 0 50 100 150 200 250 300 Figure 1. The graph of the function Q 3 (25) (a). If k is defined by the equa tion: b k ( a ) = b , then ( k − 1) /k = c 2 /d 2 . Ther efore, if | S | > 2 n + 3, then k is an in teger n umber a nd k ∈ { 2 , . . . , K ( n ) } [8]. Her e, K ( n ) = ⌊ 1+ √ 2 n 2 ⌋ . Denote by D ( n ) i,k the set o f a ll r eal num b ers a such that ( a, b k ( a )) ∈ D ( n ) i . Let R ( n ) i,k ( a ) := ( U ( n ) i ( a, b k ( a )) for a ∈ D ( n ) i,k ∞ for a / ∈ D ( n ) i,k Q ( n ) k ( a ) := min i n R ( n ) i,k ( a ) o Then Theorem 4 yields the following b ound for | S | : Theorem 5. L et S b e a spheric al two-distanc e set in R n with inner pr o du ct s a and b k ( a ) . Then | S | ≤ Q ( n ) k ( a ) . Consider the case a + b k ( a ) < 0. Since b k ( a ) ≥ − 1, we have a ∈ I k :=  2 − k k , 1 2 k − 1  . Remark 1. A ctual ly, the p olynomials P i ar e chosen su ch t hat the maximum of Q ( n ) k ( a ) on I k minimize the Delsarte b ou n d (The or em 3). Cle arly, Q ( n ) k ( a ) is a pie c ewise ra tional function on I k . It is not har d t o find explic it expr essions for Q ( n ) k ( a ) and to c ompute its maximum on I k numeric al ly. F or inst anc e, max { Q (25) 3 ( a ) : a ∈ I 3 = [ − 1 / 3 , 1 / 5) } ≈ 284 . 1 4 ( se e Fig. 1). 6 4 Maximal spherical t w o-d istance sets In this section we use Theorem 5 to b ound the cardinality of a spherical tw o- distance set with a + b < 0. Let S , | S | > 2 n + 3 , b e a spherical tw o-dis ta nce set in R n with inner pro ducts a and b k ( a ). Then k ∈ { 2 , . . . , K ( n ) } , and − 1 ≤ b k ( a ) < a < 1. Let ˜ K ( n ) := max { K ( n ) , 2 } . F or given n and k = 2 , . . . , ˜ K ( n ), we denote b y Ω( n, k ) the set of all spher ical tw o-distance sets S in R n with a + b k ( a ) < 0. Denote by ω ( n, k ) the la rgest cardinality of S ∈ Ω( n, k ). Let ϕ ( n, k ) := sup a ∈ I k n Q ( n ) k ( a ) o , b ω ( n, k ) := ma x {⌊ ϕ ( n, k ) ⌋ , 2 n + 3 } . Let us denote by b ω ( n ) the maximum of num b ers b ω ( n, 2) , . . . , b ω ( n, ˜ K ( n )), and b y ω ( n ) we denote the larges t cardinality of a tw o-dista nce set S in S n − 1 with a + b < 0. Then g ( n ) = max { ω ( n ) , ρ ( n ) } . Since Theorem 5 implies ω ( n, k ) ≤ b ω ( n, k ) , we hav e Theorem 6. g ( n ) ≤ max { b ω ( n ) , ρ ( n ) } . Finally , for g ( n ) we hav e the following bo unds: ρ ( n ) ≤ g ( n ) ≤ max { b ω ( n ) , ρ ( n ) } . Recall that ρ ( n ) = n ( n + 1) / 2 for n ≥ 7 . F or b ω ( n ) , 7 ≤ n ≤ 40 , we obta in the computational results ga ther ed in T able 1. Since b ω ( n ) ≤ ρ ( n ) for 6 < n < 40 , n 6 = 22 , 23, for these cases we ha ve g ( n ) = ρ ( n ). F or n = 23 we o btain g (23) ≤ 277. But g (23) ≥ ρ (23) = 276. This prov es the following theorem: Theorem 7. If 6 < n < 22 or 23 < n < 40 , t hen g ( n ) = n ( n + 1) 2 . F or n = 23 we have g (23) = 276 or 277 . Remark 2. The c ase n = 23 is very int er est ing. In this dimension the maximal numb er of e quiangular lines (or e quivalently, the maximal c ar dinality of a two- distanc e set with a + b = 0 ) is 276 [9]. On the other hand, | Λ 23 | = 276 . S o in 23 dimensions we have two very differ ent two-distanc e sets with 27 6 p oints. Note that ma x { Q (23) 3 ( a ) : a ∈ I 3 } ≈ 277 . 09 5 . So this nu meric al b ound is not far fr om 277 . Perhaps str onger to ols, such as semidefinite pr o gr amming b ounds, ar e ne e de d her e to pr ove that g (2 3) = 276 . Remark 3. Our numeric al c alculations show that the b arrier n = 40 is in fact fundamental: LP b ounds ar e inc ap able of r esolving t he n ≥ 4 0 , k = 2 c ase. That me ans a new ide a is r e quir e d t o de al with n ≥ 40 . 7 T able 1. b ω ( n ) and ρ ( n ). The last column gives the k with b ω ( n ) = b ω ( n, k ) . n b ω ρ k 7 28 28 2 8 31 36 2 9 34 45 2 10 37 55 2 11 40 66 2 12 44 78 2 13 47 91 2 14 52 105 2 15 56 120 2 16 61 136 2 17 66 153 2 18 76 171 3 19 96 190 3 20 12 6 210 3 21 17 6 231 3 22 27 5 253 3 23 27 7 276 3 24 28 0 300 3 25 28 4 325 3 26 28 8 351 3 27 29 4 378 3 28 29 9 406 3 29 30 5 435 3 30 31 2 465 3 31 31 9 496 3 32 32 7 528 3 33 33 4 561 3 34 34 2 595 3 35 36 0 630 2 36 41 6 666 2 37 48 8 703 2 38 58 4 741 2 39 72 1 780 2 40 92 8 820 2 Remark 4. It is known that for n = 3 , 7 , 23 maximal spheric al two-distanc e sets ar e not unique, and for n = 2 , 6 , 22 , when g ( n ) = n ( n + 3) / 2 , these sets ar e unique up to isometry. Lison ˇ ek [10] c onfirme d the maximali ty and uniqueness of pr eviously known sets for n = 4 , 5 , 6 . F or al l other n the pr oblem of u niqueness of maximal two-distanc e sets is op en. We think that for 7 < n < 4 6 , n 6 = 22 , 23 maximal spheric al two-distanc e sets in R n ar e u nique and c ongruent to Λ n . 8 Ac kno wledgmen ts The author thanks Alexander Barg and F rank V allen tin for useful dis cussions and comments. References [1] E. Ba nnai, E. Bannai, and D. Stan ton, An upper b ound for the cardinality of an s -dista nc e set in rea l Euclidean s pace, Combinatoric a , 3 (19 83), 1 4 7- 152. [2] A. Blokhuis, A new upper b o und for the cardinalit y of 2-dis tance set in Euclidean space, Ann. Discr ete Math. , 20 (1984 ), 65 -66. [3] A. Blok h uis, F ew-distance sets, CWI T ract 7 (1 984). [4] J. H. Conw ay and N. J. A. Sloane, Sphere Pac kings , Lattices, and Gr oups, Springer-V erla g, New Y ork-Ber lin, 1988 . [5] Ph. Delsarte, J. M. Goetha ls and J. J. Seidel, Spherica l co des and designs, Ge om. De dic. , 6 , 1 9 77, 3 63-388 . [6] S. J. Einhorn and I. J. Schoenber g, On Euclidean sets ha ving o nly tw o distances b et ween p oints I, II, Indag. Math. , 28 (1966), 479-4 88, 489-5 04. (Nederl. Acad. W etensch. Pro c. Ser . A69 ) [7] G. A. K abatiansky and V. I. Lev enshtein, Bounds for packings o n a spher e and in space , Pr oblems of Information T r ansmission , 14 (1), 1978, 1-17. [8] D. G. La rman, C. A. Rogers, and J. J. Seidel, On t wo-distance sets in Euclidean space, Bul l. L ondon Math. So c. , 9 (197 7), 261 -267. [9] P . W. H. Lemmens and J. J. Seidel, Equiangular lines, J. Algebr a , 24 (1973), 494 -512. [10] P . Lisonˇ ek, New maximal tw o-distance s ets, J. Comb. The ory, Ser. A , 77 (1997), 318 -338. 9