On almost-equidistant sets

ON ALMOST-EQUIDISTANT SETS A. POLYANSKII

arXiv:1707.00295v1 [math.MG] 2 Jul 2017

Abstract. A finite set V ⊂ Rd is called almost-equidistant if for any three distinct points of V there are two that are at unit distance apart. We investigate the largest cardinality of an almost-equidistant set in Rd and prove the upper bound |V| = O(d13/9 ).

1. Introduction A set S of n (unit) vectors in Rd is called almost orthogonal if among any three vectors of S there is at least one orthogonal pair. Erd˝os asked [Ros91]: what is the largest cardinality fπ/2 (d) of an almost orthogonal set of vectors in Rd ? Using an elegant linear algebraic argument Rosenfeld [Ros91] proved that fπ/2 (d) = 2d. Clearly, we can take two sets of vectors, each containing d orthogonal vectors, and obtain an almost orthogonal set of 2d vectors in Rd . Another nice proof of Rosenfeld’s theorem was given in [Dea11, Theorem 3.5]. Moreover, Deaett exhibited a different example of almost orthogonal sets in Rd of size 2d, see [Dea11, Theorem 4.11]. The following definition naturally generalize the concept of almost orthogonal sets of vectors: a finite set S of n (unit) vectors in Rd is called almost α-angular if among any three vectors of S, two of them form a fixed angle α. In particular, an almost orthogonal set is an almost π/2-angular set. The following variation of Erd˝os problem have been also considered: what is the largest cardinality fα (d) of an almost α-angular set of vectors in Rd , where α is a fixed angle? Bezdek and Lángi [BL99, Theorem 1] foundpthat fα (d) ≤ 2d + 2 for any π/2 < α < π and d ≥ 2. Obviously, for α = αd := 2 arcsin (d + 1)/(2d) one can take d + 1 unit vectors in Rd such that an angle between any two of them is αd ; note that the ends of these vectors correspond to vertices of a regular d-simplex inscribed in the unit sphere with center in the origin. Therefore, the union of two such sets of vectors is an almost αd -angular set of 2(d + 1) vectors in Rd . It is worth pointing out that the argument of Bezdek and Lángi is a natural modification of Rosenfeld’s approach. The key idea of their proof is based on two facts: the first fact is that the matrix Vα := hvi , vj i − cos α, where S = {v1 , . . . , vn } is an almost α-angular set in Rd ,

is nonnegative definite for π/2 < α < π; the second fact is that the trace of the matrix (Vα − In )3 is equal to 0, where In is the identity matrix of size n. For more details we refer the interested readers to Subsection 3.1. Since cos α > 0 for 0 < α < π/2, the matrix Vα can have one negative eigenvalue, i.e. the argument of Bezdek and Lángi does not work for small α. Unfortunately, we still do not know whether fα (d) = Oα (d) holds for any 0 < α < π/2. The current article is devoted to the generalization of almost α-angular sets. A finite set V of n points in Rd is called almost-equidistant if among any three points of V there are two points that are at unit distance apart. The following question was posed by Zsolt 2010 Mathematics Subject Classification. 51K99, 05C50, 51F99, 52C99, 05A99. Key words and phrases. Equilateral sets, almost-equilateral sets, unit distance graph, graph of diameters. A. Polyanskii, On almost-equidistant sets. 1

Lángi: what is the largest cardinality fae (d) of an almost-equidistant set of points in Rd ? Denote by G = (V, E) the graph such that its vertices are points of an almost orthogonal set V in Rd and edges are formed by pairs of edges that are at unit distance apart. Note that G does not contain a clique of size d + 2, because it is impossible to embed a regular unit (d + 1)-simplex in Rd . Moreover, it does not contain an anticlique of size 3, because its vertex set is an almost-equidistant set. By the upper bound on the Ramsey number R(3, d + 2) from the article [Gri83], we have fae (d) ≤ 2.4 · (d + 2)2 / log(d + 2). Bezdek, Nadzódi and Visy [BNV03, Statement (2) in Theorem 4] proved that fae (2) = 7; notice that the famous Moser spindle has 7 vertices forming an almost-equidistant set in the plane R2 . Recently, Balko, Pór, Scheucher, Swanepoel and Valtr [BPS+ 16] showed that fae (d) = O(d3/2 ). For that, they used the idea that appeared in Deaet’s proof [Dea11] of the equality fπ/2 (d) = 2d. Moreover, they constructed an example of an almostequidistant set in Rd with 2d + 4 vertices for d ≥ 3 and proved some bounds for the largest cardinalities of almost-equidistant set in Rd , d ≤ 9. The main result of the article is the following improvement. Theorem 1. fae (d) ≤ 5d13/9 . Obviously, we obtain the following corollary. Corollary 2. fα (d) ≤ 5d13/9 . The article is organized in the following way. Section 2 presents some preliminaries, in particular we discuss some properties of distance matrices and unit distance graphs formed by vertices of almost-equidistant sets. In Section 3 we prove Theorem 1. In Section 4 we will look at some open problems related to almost-equidistant sets. Along the way, we obtain some facts (Lemma 3 and Lemma 8) that are useful in studying distance graphs in Rd , for example, unit distance graphs and diameter graphs. 2. Preliminaries 2.1. Properties of the distance matrix. Suppose that V = {v1 , . . . , vn } ⊂ Rd is an arbitrary set of distinct points, where n ≥ 2. Let us consider the following matrix V := kvi − vj k2 .

Lemma 3. The matrix V has one positive eigenvalue. Proof. Note that V is not the zero matrix of size n. Thus it has an eigenvalue different from 0. Moreover, tr(V) = 0, i.e. the matrix V has one positive eigenvalue. Let us prove that V has at most one positive eigenvalue. Clearly, where

V = v t j + j t v − 2hvi , vj i,

v = (kv1 k2 , . . . , kvn k2 ) and j = (1, . . . , 1). It is easy to check that the eigenvalues of v t j + j t v should satisfy the following equality det(v t j + j t v − µIn ) = n X X 2 n n−2 2 = (−1) µ µ − 2 kvi k2 µ − kvi k2 − kvj k2 = 0. i=1

1≤i 1. We write λ1 , . . . , λk for eigenvalues that are less then 1. By Lemma 6, we have k ≤ d + 1 and λ+

k X i=1

3

λ +

k X i=1

λi + (n − k − 1) = 0, λ3i + (n − k − 1) = 0.

Without loss of generality assume that λ1 , . . . , λl ≤ 0 and λl+1 , . . . , λk > 0, where 1 ≤ l ≤ k. Therefore, we have l X i=1

λ3 =

l X

(−λi ) = λ + (n − k − 1) +

(−λi )3 +

i=1

k X

k X

i=l+1

(−λi )3 − (n − k − 1) ≥

i=l+1

≥

λi ≥ n + λ − d − 2 ≥ n − d − 1, (−λ1 − . . . − λl )3 − (k − l) − (n − k − 1) ≥ l2

(n − d − 1)3 − n. (d + 1)2

In order to prove that n ≤ 5d13/9 assume the contrary n = 5dβ ≥ 4dβ + d + 1 for some β ≥ 13/9, thus 64d3β − 5dβ ≥ 11d3β−2 , i.e. λ ≥ 2dβ−2/3 . λ3 ≥ 4d2 We need Gershgorin circle theorem [Ger31] (or [Pra94, Problem 34.1]). Theorem 11 (Gershgorin circle theorem). Every eigenvalue of a matrix A = aij over C of size n × n belong to one of the disks ( ) X z ∈ C : |akk − z| ≤ ρk , where ρk = |akj | for k = 1, . . . , n. 1≤j≤n,j6=k

Using Gershgorin circle theorem for the matrix U we can assume n X kv1 − vj k2 − 1 ≥ λ ≥ 2dβ−2/3 . j=2

By Lemma 7, on the left hand side there are at least n − d − 2 terms equal to 0. There is no loss of generality in assuming d+2 X kv1 − vj k2 − 1 ≥ 2dβ−2/3 . j=2

Therefore, without loss of generality we have [(d+1)/2]+1 X 2 kv1 − vj k − 1 ≥ dβ−2/3 j=2 6

Denote by α the number such that α/2 = β − 2/3 − 1 + α, i.e. α = 10/3 − 2β. Thus, we can certainly assume that α [dX ]+1 2 kv1 − vj k − 1 ≥ dα/2 . j=2

Now, we are ready to finish the proof of Theorem 1 in the spirit of the proof of the theorem saying that fae (d) = O(d3/2 ) (see [BPS+ 16]). By Lemma 8, the number of vertices in G that are adjacent to all vi for 1 ≤ i ≤ [dα ] + 1 is at most 2d + 2. By Lemma 7, the number of vertices that are not adjacent to all vi for 1 ≤ i ≤ [dα ] + 1 is at most (dα + 1)(d + 1). Therefore, n ≤ 2(d + 2) + (dα + 1)(d + 1) < 5d1+α . Since α = 10/3 − 2β ≤ 10/3 − 26/9 = 4/9, we get n < 5d13/9 , a contradiction. Theorem 1 is proved. Proof of Corollary 2. Suppose that S = {u1 , . . . , un } is an almost α-angular set of unit ˆ n } is an almost-equidistant set of points in vectors in Rd . Clearly, the set {ˆ u1 , . . . , u d ˆ i = ui /(2 sin(α/2)) for 1 ≤ i ≤ n. Therefore, n ≤ 5d13/9 . Corollary 2 is R , where u proved. 4. Discussion Unfortunately, we are not capable to prove the following natural conjecture. Conjecture 12. fae (d) = O(d).

√ Clearly, if the diameter of an almost-equidistant set V in Rd is√at least 2 then we can easily prove that |V| ≤ 4d + 6. Indeed, suppose kv − uk ≥ 2 for v, u ∈ V. By Lemma 8, the number of points in V that are at unit distance from v and u is at most 2d + 2. By Lemma 7, the number of points in V that are not at unit distance from u or v is at most 2d + 2. Therefore, |V| ≤ 4d + 6. It means that the most interesting case √ of Conjecture 12 is when the diameter of an almost-equidistant set is less or equal 2. Therefore, it would be natural to ask the following question. Problem 13. A subset of Rd is called an almost-equidistant diameter set if it is an almost-equidistant set in Rd and has diameter 1. What is the largest cardinality of an almost-equidistant diameter set in Rd ? Acknowledgement We are grateful to Zsolt Lángi for posing his question about the largest cardinality of almost-equidistant sets, to Konrad Swanepoel for several interesting and stimulating discussions on the problem. References [BL99] K. Bezdek and Z. L´ angi, Almost equidistant points on S d−1 , Period. Math. Hungar. 39 (1999), no. 1-3, 139–144. Discrete geometry and rigidity (Budapest, 1999). [BNV03] K. Bezdek, M. Naszódi, and B. Visy, On the mth Petty numbers of normed spaces, Discrete geometry, 2003, pp. 291–304. [BPS+ 16] M. Balko, A. P´ or, M. Scheucher, K. Swanepoel, and P. Valtr, Almost-equidistant sets, 2016. arXiv:1706.06375. [Dea11] L. Deaett, The minimum semidefinite rank of a triangle-free graph, Linear Algebra Appl. 434 (2011), no. 8, 1945–1955. 7

[DM94] M. Deza and H. Maehara, A few applications of negative-type inequalities, Graphs Combin. 10 (1994), no. 3, 255–262. ¨ [Ger31] S. Gerˇsgorin, Uber die Abgrenzung der Eigenwerte einer Matrix, Bulletin de l’Académie des Sciences de l’URSS 6 (1931), 749–754. [Gri83] J. R. Griggs, An upper bound on the Ramsey numbers R(3, k), J. Combin. Theory Ser. A 35 (1983), no. 2, 145–153. [Pra94] V. V. Prasolov, Problems and theorems in linear algebra, Translations of Mathematical Monographs, vol. 134, American Mathematical Society, Providence, RI, 1994. [Ros91] M. Rosenfeld, Almost orthogonal lines in E d , Applied geometry and discrete mathematics, 1991, pp. 489–492. [Wey12] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912), no. 4, 441–479. Alexandr Polyanskii Moscow Institute of Physics and Technology Institutskiy per. 9 Dolgoprudny, Russia 141700 Technion – Israel Institute of Technology Haifa, Israel 32000 Institute for Information Transmission Problems RAS Bolshoy Karetny per. 19 Moscow, Russia 127994 E-mail address: [email protected]

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