{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T21:17:16Z","timestamp":1776374236782,"version":"3.51.2"},"reference-count":17,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T00:00:00Z","timestamp":1680307200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,7,21]],"date-time":"2023-07-21T00:00:00Z","timestamp":1689897600000},"content-version":"vor","delay-in-days":111,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100006919","name":"Massachusetts Institute of Technology","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100006919","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2023,4]]},"abstract":"Abstract<\/jats:title>We study the problem of determining the maximum size of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Let $$N_{\\alpha ,\\beta }(d)$$<\/jats:tex-math>\n \n \n N<\/mml:mi>\n \n \u03b1<\/mml:mi>\n ,<\/mml:mo>\n \u03b2<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> denote the maximum number of unit vectors in $${\\mathbb {R}}^d$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> where all pairwise inner products lie in $$\\{\\alpha ,\\beta \\}$$<\/jats:tex-math>\n \n {<\/mml:mo>\n \u03b1<\/mml:mi>\n ,<\/mml:mo>\n \u03b2<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. For fixed $$-1\\le \\beta<0\\le \\alpha <1$$<\/jats:tex-math>\n \n -<\/mml:mo>\n 1<\/mml:mn>\n \u2264<\/mml:mo>\n \u03b2<\/mml:mi>\n <<\/mml:mo>\n 0<\/mml:mn>\n \u2264<\/mml:mo>\n \u03b1<\/mml:mi>\n <<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we propose a conjecture for the limit of $$N_{\\alpha ,\\beta }(d)\/d$$<\/jats:tex-math>\n \n \n N<\/mml:mi>\n \n \u03b1<\/mml:mi>\n ,<\/mml:mo>\n \u03b2<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \/<\/mml:mo>\n d<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> as $$d \\rightarrow \\infty $$<\/jats:tex-math>\n \n d<\/mml:mi>\n \u2192<\/mml:mo>\n \u221e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in terms of eigenvalue multiplicities of signed graphs. We determine this limit when $$\\alpha +2\\beta <0$$<\/jats:tex-math>\n \n \u03b1<\/mml:mi>\n +<\/mml:mo>\n 2<\/mml:mn>\n \u03b2<\/mml:mi>\n <<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> or $$(1-\\alpha )\/(\\alpha -\\beta ) \\in \\{1, \\sqrt{2}, \\sqrt{3}\\}$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n 1<\/mml:mn>\n -<\/mml:mo>\n \u03b1<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \/<\/mml:mo>\n \n (<\/mml:mo>\n \u03b1<\/mml:mi>\n -<\/mml:mo>\n \u03b2<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2208<\/mml:mo>\n \n {<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \n 2<\/mml:mn>\n <\/mml:msqrt>\n ,<\/mml:mo>\n \n 3<\/mml:mn>\n <\/mml:msqrt>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>Our work builds on our recent resolution of the problem in the case of $$\\alpha = -\\beta $$<\/jats:tex-math>\n \n \u03b1<\/mml:mi>\n =<\/mml:mo>\n -<\/mml:mo>\n \u03b2<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (corresponding to equiangular lines). It is the first determination of $$\\lim _{d \\rightarrow \\infty } N_{\\alpha ,\\beta }(d)\/d$$<\/jats:tex-math>\n \n \n lim<\/mml:mo>\n \n d<\/mml:mi>\n \u2192<\/mml:mo>\n \u221e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n N<\/mml:mi>\n \n \u03b1<\/mml:mi>\n ,<\/mml:mo>\n \u03b2<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \/<\/mml:mo>\n d<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for any nontrivial fixed values of $$\\alpha $$<\/jats:tex-math>\n \u03b1<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\beta $$<\/jats:tex-math>\n \u03b2<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> outside of the equiangular lines setting.\n<\/jats:p>","DOI":"10.1007\/s00493-023-00002-1","type":"journal-article","created":{"date-parts":[[2023,7,21]],"date-time":"2023-07-21T08:03:23Z","timestamp":1689926603000},"page":"203-232","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Spherical Two-Distance Sets and Eigenvalues of Signed Graphs"],"prefix":"10.1007","volume":"43","author":[{"given":"Zilin","family":"Jiang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jonathan","family":"Tidor","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuan","family":"Yao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shengtong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yufei","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,7,21]]},"reference":[{"key":"2_CR1","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1007\/s00222-017-0746-0","volume":"211","author":"I Balla","year":"2018","unstructured":"Balla, I., Dr\u00e4xler, F., Keevash, P., Sudakov, B.: Equiangular lines and spherical codes in Euclidean space. 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