## Equiangular lines with a fixed angle.(English)Zbl 1478.52015

Summary: Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle.
Fix $$0<\alpha<1$$. Let $$N_\alpha (d)$$ denote the maximum number of lines through the origin in $$\mathbb{R}^d$$ with pairwise common angle $$\operatorname{arccos}\alpha$$. Let $$k$$ denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly $$(1-\alpha)/(2\alpha)$$. If $$k<\infty$$, then $$N_\alpha(d)=\lfloor k(d-1)/(k-1)\rfloor$$ for all sufficiently large $$d$$, and otherwise $$N_\alpha (d)=d+o(d)$$. In particular, $$N_{1/(2k-1)}(d)=\lfloor k(d-1)/(k-1)\rfloor$$ for every integer $$k\ge 2$$ and all sufficiently large $$d$$.
A key ingredient is a new result in spectral graph theory: the adjacency matrix of a connected bounded degree graph has sublinear second eigenvalue multiplicity.

### MSC:

 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry) 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
Full Text:

### References:

  Balla, Igor; Dr\"{a}xler, Felix; Keevash, Peter; Sudakov, Benny, Equiangular lines and spherical codes in {E}uclidean space, Invent. Math.. Inventiones Mathematicae, 211, 179-212 (2018) · Zbl 1383.51035  Barg, Alexander; Yu, Wei-Hsuan, New bounds for equiangular lines. Discrete geometry and algebraic combinatorics, Contemp. Math., 625, 111-121 (2014) · Zbl 1333.52027  Bukh, Boris, Bounds on equiangular lines and on related spherical codes, SIAM J. Discrete Math.. SIAM Journal on Discrete Mathematics, 30, 549-554 (2016) · Zbl 1333.05309  de Caen, D., Large equiangular sets of lines in {E}uclidean space, Electron. J. Combin.. Electronic Journal of Combinatorics, 7, 55-3 (2000) · Zbl 0966.51010  Colding, Tobias H.; Minicozzi, II, William P., Harmonic functions on manifolds, Ann. of Math. (2). Annals of Mathematics. Second Series, 146, 725-747 (1997) · Zbl 0928.53030  Davidoff, Giuliana; Sarnak, Peter; Valette, Alain, Elementary Number Theory, Group Theory, and {R}amanujan Graphs, London Math. Soc. Stud. Texts, 55, x+144 pp. (2003) · Zbl 1032.11001  Glazyrin, Alexey; Yu, Wei-Hsuan, Upper bounds for {$$s$$}-distance sets and equiangular lines, Adv. Math.. Advances in Mathematics, 330, 810-833 (2018) · Zbl 1394.52026  Godsil, Chris; Royle, Gordon, Algebraic Graph Theory, Grad. Texts in Math., 207, xx+439 pp. (2001) · Zbl 0968.05002  Gowers, W. T., Quasirandom groups, Combin. Probab. Comput.. Combinatorics, Probability and Computing, 17, 363-387 (2008) · Zbl 1191.20016  Greaves, Gary; Koolen, Jacobus H.; Munemasa, Akihiro; Sz\"{o}ll\H{o}si, Ferenc, Equiangular lines in {E}uclidean spaces, J. Combin. Theory Ser. A. Journal of Combinatorial Theory. Series A, 138, 208-235 (2016) · Zbl 1330.51006  Gromov, Mikhael, Groups of polynomial growth and expanding maps, Inst. Hautes \'{E}tudes Sci. Publ. Math.. Institut des Hautes \'{E}tudes Scientifiques. Publications Math\'{e}matiques, 53, 53-73 (1981) · Zbl 0474.20018  Haantjes, J., Equilateral point-sets in elliptic two- and three-dimensional spaces, Nieuw Arch. Wiskunde (2), 22, 355-362 (1948) · Zbl 0037.21703  Jiang, Zilin; Polyanskii, Alexandr, Forbidden subgraphs for graphs of bounded spectral radius, with applications to equiangular lines, Israel J. Math.. Israel Journal of Mathematics, 236, 393-421 (2020) · Zbl 1439.05138  Jiang, Zilin; Tidor, Jonathan; Yao, Yuan; Zhang, Shengtong; Zhao, Yufei, Spherical two-distance sets and eigenvalues of signed graphs (2020)  Kleiner, Bruce, A new proof of {G}romov’s theorem on groups of polynomial growth, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 23, 815-829 (2010) · Zbl 1246.20038  Lee, James R.; Makarychev, Yury, Eigenvalue multiplicity and volume growth (2008)  Lemmens, P. W. H.; Seidel, J. J., Equiangular lines, J. Algebra. Journal of Algebra, 24, 494-512 (1973) · Zbl 0255.50005  van Lint, J. H.; Seidel, J. J., Equilateral point sets in elliptic geometry, Nederl. Akad. Wetensch. Proc. Ser. A, 69, 335-348 (1966) · Zbl 0138.41702  Neumaier, A., Graph representations, two-distance sets, and equiangular lines, Linear Algebra Appl.. Linear Algebra and its Applications, 114/115, 141-156 (1989) · Zbl 0724.05043  Renes, Joseph M.; Blume-Kohout, Robin; Scott, A. J.; Caves, Carlton M., Symmetric informationally complete quantum measurements, J. Math. Phys.. Journal of Mathematical Physics, 45, 2171-2180 (2004) · Zbl 1071.81015  Strohmer, Thomas; Heath, Jr., Robert W., Grassmannian frames with applications to coding and communication, Appl. Comput. Harmon. Anal.. Applied and Computational Harmonic Analysis. Time-Frequency and Time-Scale Analysis, Wavelets, Numerical Algorithms, and Applications, 14, 257-275 (2003) · Zbl 1028.42020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.