arXiv:0801.3706v1 [math.MG] 24 Jan 2008

Spherical two-distance sets Oleg R. Musin

∗

Abstract A set S of unit vectors in n−dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b, and inner products of distinct vectors of S are either a or b. The largest cardinality g(n) of spherical two-distance sets is not exceed n(n + 3)/2. This upper bound is tight for n = 2, 6, 22. The set of mid-points of the edges of a regular simplex gives for g(n) a lower bound L(n) = n(n + 1)/2. In this paper using so-called (independent) polynomials’ method is proved that for nonnegative a+b the largest cardinality of S is not greater than L(n). For the case a + b < 0 we propose upper bounds 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.

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Introduction

A set S in Euclidean space Rn is called a two-distance set, if there are two distances c and d, and the distances between pairs of points of S are either c or d. If a two-distance set S lies in the unit sphere Sn−1 , then S is called spherical two-distance set. In other words, S is a set of unit vectors, there are two real numbers a and b, −1 ≤ a, b < 1, and inner products of distinct vectors of S are either a or b. The ratios of distances of two-distance sets are quit restrictive. Namely, Larman, Rogers, and Seidel [8] have proved the following fact: if the cardinality of a two-distance set S in Rn , with distances c and d, c < d, is greater than 2n + 3, then the ratio c2 /d 2 = (k − 1)/k, where k is an integer number, and √ 1 + 2n . 2≤k≤ 2 Einhorn and Schoenberg [6] proved that there are finitely many two-distance sets S in Rn of the cardinality |S| ≥ n + 2. Delsarte, Goethals, and Seidel [5] proved that the largest cardinality of spherical two-distance sets in Rn (we denote it by g(n)) is bounded by n(n + 3)/2, i.e., g(n) ≤ ∗ Department

n(n + 3) . 2

of Mathematics, University of Texas at Brownsville. [email protected]

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Moreover, they give examples of spherical two-distance sets with n(n + 3)/2 points for n = 2, 6, 22.1 Blockhuis [2] shows that the cardinality of (Euclidean) two-distance sets in Rn is not exceed (n + 1)(n + 2)/2. Let unit vectors e1 , . . . , en+1 form an orthogonal basis of Rn+1 . Denote by ∆n the regular simplex with vertices 2e1 , . . . , 2en+1 . Let Λn bePthe set of points ei + ej , 1 ≤ i < j ≤ n + 1. Since Λn lies in the hyperplane xk = 2, we see that Λn represents a spherical two-distance set in Rn . On the other hand, Λn is the set of mid-points of the edges of ∆n . Thus, g(n) ≥ |Λn | =

n(n + 1) . 2

For n < 7 the largest cardinality of Euclidean two-distance sets is g(n), where g(2) = 5, g(3) = 6, g(4) = 10, g(5) = 16, and g(6) = 27 (see [10]). However, for n = 7, 8 Lisonˇek [10] discovered non-spherical maximal two-distance sets of the cardinality 29 and 45 respectively. In this paper we prove that g(n) =

n(n + 1) , where 6 < n < 40, n 6= 22, 23, 2


and g(23) = 276 or 277. This proof (Section 4) is based on the bound n+1 2 for spherical two-distance sets with a + b ≥ 0 (Section 2), and on the Delsarte bounds for spherical two-distance sets in the case a + b < 0 (Section 3).

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Linearly independent polynomials


The upper bound n(n + 3)/2 for spherical two-distance sets [5], the bound n+2 2
for Euclidean two-distance sets [2], as well as the bound n+s for s−distance s sets [1, 3] were obtained by the (linearly independent) polynomials method. The main idea of this method is the following: to associate sets to some polynomials and show that these polynomials are linearly independent as members of the corresponding vector space. Now we apply this idea to improve upper bounds for spherical two-distance sets with a + b ≥ 0. Theorem 1. Let S be a spherical two-distance set in Rn with inner products a and b. If a + b ≥ 0, then n(n + 1) |S| ≤ . 2 Proof. Let (t − a)(t − b) . F (t) := (1 − a)(1 − b) For any unit vector y ∈ Rn we define the function Fy : Sn−1 → R by x ∈ Rn , ||x|| = 1.

Fy (x) := F (hx, yi), 1 Therefore,

in these dimensions we have equalities g(n) = n(n + 3)/2.

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Let S = {x1 , . . . , xm }, i.e. |S| = m. Denote fi (x) := Fxi (x). Since fi (xj ) = δi,j ,

(1)

the quadratic polynomials fi , i = 1, . . . , m, are linearly independent. Let e1 , . . . , en be a basis of Rn . Let Li (x) := hx, ei i, x ∈ Sn−1 . Then the linear polynomials L1 , . . . , Ln are also linearly independent. Now we show that if a + b ≥ 0, then f1 , . . . , fm , L1 , . . . , Ln form a linearly independent system of polynomials. Indeed, assume the converse. Then n X

dk Lk (x) =

m X

ci fi (x),

i=1

k=1

where there are nonzero dk and ci . Let v = d1 e1 + . . . + dn en . Then hx, vi = For x = xi in (2), using (1), we get

X

ci fi (x).

(2)

i

ci = hxi , vi. Take x = v and x = −v in (2). Then we have X X ci F (ci ), ci fi (v) = ||v||2 = −||v||2 =

X

ci fi (−v) =

X

ci F (−ci ).

(4)

i

i

Subtracting (3) from (4), we obtain −||v||2 =

(3)

i

i

X a+b c2 . (1 − a)(1 − b) i i

That yields v = 0, a contradiction. Note that the dimension of the vector space of quadratic polynomials on the sphere Sn−1 is n(n + 3)/2. Therefore, dim {f1 , . . . , fm , L1 , . . . , Ln } = m + n ≤

n(n + 3) . 2

Thus, |S| = m ≤ n(n + 1)/2. Denote by ρ(n) the largest possible cardinality of spherical two-distance sets in Rn with a + b ≥ 0. 3

Theorem 2. If n ≥ 7, then ρ(n) =

n(n + 1) 2

Proof. Theorem 1 implies that ρ(n) ≤ n(n + 1)/2. On the other hand, the set of mid-points of the edges of a regular simplex has n(n + 1)/2 points and a + b ≥ 0 for n ≥ 7. Indeed, for this spherical two-distance set we have a=

n−3 , 2(n − 1)

Thus, a+b=

3

b=

−2 . n−1

n−7 ≥ 0. 2(n − 1)

Delsarte’s method for two-distance sets

Delsarte’s method is widely used in coding theory and discrete geometry for finding bounds for error-correcting codes, spherical codes, and sphere packings (see [4, 5, 7]). In this method upper bounds for spherical codes are given by the following theorem: Theorem 3 ([5, 7]). Let T be a subset of the interval [−1, 1]. Let S be a set of unit vectors in Rn such that the set of inner products of distinct vectors of S lies in T . Suppose a polynomial f is a nonnegative linear combination of (n) Gegenbauer polynomials Gk (t), i.e., X (n) fk Gk (t), where fk ≥ 0. f (t) = k

If f (t) ≤ 0 for all t ∈ T and f0 > 0, then |S| ≤

f (1) f0

There are many ways to define Gegenbauer (or ultraspherical) polynomials (n) (α,β) Gk are a special case of Jacobi polynomials Pk with α = β = (n) (n) (n − 3)/2 and the normalization by Gk (1) = 1. Also Gk can be defined by the recurrence formula: (n) Gk (t).

(n)

(n)

G0

(n)

= 1, G1

(n)

= t, . . . , Gk

=

(n)

(2k + n − 4) t Gk−1 − (k − 1) Gk−2 . k+n−3

For instance, (n)

G2 (t) = 4

nt2 − 1 , n−1

(n)

G3 (t) =

(n + 2)t3 − 3t , n−1

(n + 2)(n + 4)t4 − 6(n + 2)t2 + 3 . n2 − 1 Now for given n, a, b we introduce polynomials Pi (t), i = 1, . . . , 5. (n)

G4 (t) =

i = 1 : P1 (t) = (t − a)(t − b).

(n)

(n)

i = 2 : P2 (t) = (t − a)(t − b)(t + c) = f0 + f1 t + f2 G2 (t) + f3 G3 (t), where c is defined by the equation f1 = 0. Then c=

ab(n + 2) + 3 , (n + 2)(a + b)

a + b 6= 0. (n)

(n)

i = 3 : P3 (t) = (t − a)(t − b)(t + a + b) = f0 + f1 t + f2 G2 (t) + f3 G3 (t). Note that f2 = 0. P (n) i = 4 : P4 (t) = (t − a)(t − b)(t2 + c t + d) = fk Gk (t), where c and d are defined by the equations f1 = f2 = 0. P (n) i = 5 : P5 (t) = (t − a)(t − b)(t2 + c t + d) = fk Gk (t), where c and d are defined by the equations f2 = f3 = 0. P (i) (n) (n) Let Pi (t) = fk Gk (t). Denote by Di the set of all pairs (a, b) such (i) (i) that the polynomial Pi (t) is well defined, all fk ≥ 0, and f0 > 0. Let (n)

Ui (a, b) :=

Pi (1) (i)

f0

.

Note that for T = {a, b} we have Pi (t) = 0 for all t ∈ T . Then Theorem 3 yields Theorem 4. Let S be a spherical two-distance set in Rn with inner products a (n) and b. Suppose (a, b) ∈ Di for some i, 1 ≤ i ≤ 5. Then (n)

|S| ≤ Ui (a, b). n Let S be a spherical √ set in R with inner products a and b, √ two-distance where a > b. Let c = 2 − 2a, d = 2 − 2b. Then c and d are distances of S. Let ka − 1 bk (a) := . k−1

If k is defined by the equation: bk (a) = b, then (k −1)/k = c2 /d 2 . Therefore, if |S| > 2n + 3, then k is an integer number and k ∈ {2, . . . , K(n)} [8]. Here by K(n) is denoted the maximal integer number such that √ 1 + 2n K(n) ≤ . 2 5

(n)

(n)

Denote by Di,k the set of all real numbers a such that (a, bk (a)) ∈ Di . Let ( (n) (n) Ui (a, bk (a)) for a ∈ Di,k (n) Ri,k (a) := (n) ∞ for a ∈ / Di,k n o (n) (n) Qk (a) := min Ri,k (a) i

Then Theorem 4 yields the following bound for |S|:

Theorem 5. Let S be a spherical two-distance set in Rn with inner products a and bk (a). Then (n) |S| ≤ Qk (a).

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Maximal spherical two-distance sets

In this section we are using Theorem 5 for bounding of |S| with a + b < 0. Let S, |S| > 2n+3, be a spherical two-distance set in Rn with inner products a and bk (a). Then k ∈ {2, . . . , K(n)}, and −1 ≤ bk (a) < a < 1. ˜ ˜ Let K(n) := max{K(n), 2}. For given n and k = 2, . . . , K(n),, we denote by n Ω(n, k) the set of all spherical two-distance sets S in R with inner products a, bk (a), and a + bk (a) < 0. Denote by ω(n, k) the largest cardinality of S ∈ Ω(n, k). If S ∈ Ω(n, k), then a + bk (a) < 0 and bk (a) ≥ −1. So then   1 2−k . , a ∈ Ik := k 2k − 1 Let

n o (n) ϕ(n, k) := sup Qk (a) , a∈Ik

ω b (n, k) := max {the integer part of ϕ(n, k), 2n + 3}.

˜ Let us denote by ω b (n) the maximum of numbers ω b (n, 2), . . . , ω b (n, K(n)), and by ω(n) we denote the largest cardinality of a two-distance set S in Sn−1 with a + b < 0. Then g(n) = max{ω(n), ρ(n)}. Since Theorem 5 implies ω(n, k) ≤ ω b (n, k),

we have

Theorem 6. g(n) ≤ max{b ω(n), ρ(n)}. Finally, for g(n) we have bounds: ρ(n) ≤ g(n) ≤ max{b ω(n), ρ(n)}. Recall that ρ(n) = n(n + 1)/2 for n ≥ 7. For ω b (n), 7 ≤ n ≤ 40, we obtain the computational results gathered in Table 1.2 2 Note

(n)

(n)

that Di,k is a semi-algebraic set in R, and Qk (a) is a piecewise rational function. So all computations can be done using any computer algebra system.

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Table 1. ω b (n) and ρ(n). The last column gives k : ω b (n) = ω b (n, k). n 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

ω b 28 31 34 37 40 44 47 52 56 61 66 76 96 126 176 275 277 280 284 288 294 299 305 312 319 327 334 342 360 416 488 584 721 928

ρ 28 36 45 55 66 78 91 105 120 136 153 171 190 210 231 253 276 300 325 351 378 406 435 465 496 528 561 595 630 666 703 741 780 820

k 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2

Since ω b (n) ≤ ρ(n) for 6 < n < 40, n 6= 22, 23, we have for these cases g(n) = ρ(n). For n = 23 3 we obtain g(23) ≤ 277. But g(23) ≥ ρ(23) = 276. It proves the following theorem: 3 The case n = 23 is very interesting. In this dimension the maximal number of equiangular lines (or equivalently, the maximal cardinality of a two-distance set with a + b = 0) is 276 [9]. On the other hand, |Λ23 | = 276. So in 23 dimensions we have at least two different two-distance sets with 276 points.

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Theorem 7. If 6 < n < 22 or 23 < n < 40, then g(n) =

n(n + 1) . 2

For n = 23 we have g(23) = 276 or 277.

References [1] E. Bannai, E. Bannai, and D. Stanton, An upper bound for the cardinality of an s-distance set in real Euclidean space, Combinatorica, 3 (1983), 147152. [2] A. Blokhius, A new upper bound for the cardinality of 2-distance set in Euclidean space, Ann. Discrete Math., 20 (1984), 65-66. [3] A. Blokhius, Few-distance sets, CWI Tract 7 (1984). [4] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups, Springer-Verlag, New York-Berlin, 1988. [5] Ph. Delsarte, J. M. Goethals and J. J. Seidel, Spherical codes and designs, Geom. Dedic., 6, 1977, 363-388. [6] S. J. Einhorn and I. J. Schoenberg, On Euclidean sets having only two distances between points I, II, Indag. Math., 28 (1966), 479-488, 489-504. (Nederl. Acad. Wetensch. Proc. Ser. A69) [7] G. A. Kabatiansky and V. I. Levenshtein, Bounds for packings on a sphere and in space, Problems of Information Transmission, 14(1), 1978, 1-17. [8] D. G. Larman, C. A. Rogers, and J. J. Seidel, On two-distance sets in Euclidean space, Bull. London Math. Soc., 9 (1977), 261-267. [9] P. W. H. Lemmens and J. J. Seidel, Equingular lines, J. Algebra, 24 (1973), 494-512. [10] P. Lisonˇek, New maximal two-distance sets, J. Comb. Theory, Ser. A, 77 (1997), 318-338.

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