{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T05:49:24Z","timestamp":1782280164077,"version":"3.54.5"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2016,6,8]],"date-time":"2016-06-08T00:00:00Z","timestamp":1465344000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2016,9]]},"DOI":"10.1007\/s00454-016-9783-5","type":"journal-article","created":{"date-parts":[[2016,6,7]],"date-time":"2016-06-07T23:36:22Z","timestamp":1465342582000},"page":"337-356","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Bisector Energy and Few Distinct Distances"],"prefix":"10.1007","volume":"56","author":[{"given":"Ben","family":"Lund","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Adam","family":"Sheffer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Frank","family":"de Zeeuw","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2016,6,8]]},"reference":[{"key":"9783_CR1","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1007\/BF02579184","volume":"3","author":"J Beck","year":"1983","unstructured":"Beck, J.: On the lattice property of the plane and some problems of Dirac, Motzkin, and Erd\u0151s in combinatorial geometry. Combinatorica 3, 281\u2013297 (1983)","journal-title":"Combinatorica"},{"key":"9783_CR2","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-9967-7","volume-title":"Graph Theory: An Introductory Course","author":"B Bollob\u00e1s","year":"1979","unstructured":"Bollob\u00e1s, B.: Graph Theory: An Introductory Course. Springer, New York (1979)"},{"key":"9783_CR3","volume-title":"Real Algebraic Geometry","author":"J Bochnak","year":"2013","unstructured":"Bochnak, J., Coste, M., Roy, M.F.: Real Algebraic Geometry. Springer, Berlin (2013)"},{"key":"9783_CR4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/jcta.1999.2976","volume":"89","author":"G Elekes","year":"2000","unstructured":"Elekes, G., R\u00f3nyai, L.: A combinatorial problem on polynomials and rational functions. J. Comb. Theory, Ser. A 89, 1\u201320 (2000)","journal-title":"J. Comb. Theory, Ser. A"},{"key":"9783_CR5","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1007\/s00493-012-2505-6","volume":"32","author":"G Elekes","year":"2012","unstructured":"Elekes, G., Szab\u00f3, E.: How to find groups? (And how to use them in Erd\u0151s geometry?). Combinatorica 32, 537\u2013571 (2012)","journal-title":"Combinatorica"},{"key":"9783_CR6","unstructured":"Elekes, G., Szab\u00f3, E.: On triple lines and cubic curves: the Orchard Problem revisited. http:\/\/arxiv.org\/abs\/1302.5777"},{"key":"9783_CR7","doi-asserted-by":"crossref","first-page":"248","DOI":"10.2307\/2305092","volume":"53","author":"P Erd\u0151s","year":"1946","unstructured":"Erd\u0151s, P.: On sets of distances of $$n$$ n points. Am. Math. Mon. 53, 248\u2013250 (1946)","journal-title":"Am. Math. Mon."},{"key":"9783_CR8","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/BF02414146","volume":"103","author":"P Erd\u0151s","year":"1975","unstructured":"Erd\u0151s, P.: On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. 103, 99\u2013108 (1975)","journal-title":"Ann. Mat. Pura Appl."},{"key":"9783_CR9","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/0012-365X(86)90009-9","volume":"60","author":"P Erd\u0151s","year":"1986","unstructured":"Erd\u0151s, P.: On some metric and combinatorial geometric problems. Discrete Math. 60, 147\u2013153 (1986)","journal-title":"Discrete Math."},{"key":"9783_CR10","unstructured":"Erd\u0151s, P., Purdy, G.: Some extremal problems in geometry IV. In: Proceedings of 7th Southeastern Conference on Combinatorics, Graph Theory, and Computing, pp. 307\u2013322 (1976)"},{"key":"9783_CR11","unstructured":"Fox, J., Pach, J., Sheffer, A., Suk, A., Zahl, J.: A semi-algebraic version of Zarankiewicz\u2019s problem. J. Eur. Math. Soc. (to appear)"},{"key":"9783_CR12","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1007\/s00454-013-9518-9","volume":"50","author":"B Green","year":"2013","unstructured":"Green, B., Tao, T.: On sets defining few ordinary lines. Discrete Comput. Geom. 50, 409\u2013468 (2013)","journal-title":"Discrete Comput. Geom."},{"key":"9783_CR13","doi-asserted-by":"crossref","first-page":"155","DOI":"10.4007\/annals.2015.181.1.2","volume":"181","author":"L Guth","year":"2015","unstructured":"Guth, L., Katz, N.H.: On the Erd\u0151s distinct distances problem in the plane. Ann. Math. 181, 155\u2013190 (2015)","journal-title":"Ann. Math."},{"key":"9783_CR14","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1016\/j.ffa.2015.10.002","volume":"37","author":"B Hanson","year":"2016","unstructured":"Hanson, B., Lund, B., Roche-Newton, O.: On distinct perpendicular bisectors and pinned distances in finite fields. Finite Fields Appl. 37, 240\u2013264 (2016)","journal-title":"Finite Fields Appl."},{"key":"9783_CR15","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1090\/conm\/342\/06136","volume":"342","author":"NH Katz","year":"2004","unstructured":"Katz, N.H., Tardos, G.: A new entropy inequality for the Erd\u0151s distance problem, Towards a Theory of Geometric Graphs (J. Pach, ed.). Contemp. Math. 342, 119\u2013126 (2004)","journal-title":"Contemp. Math."},{"key":"9783_CR16","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1007\/s003730200063","volume":"18","author":"J Pach","year":"2002","unstructured":"Pach, J., Tardos, G.: Isosceles triangles determined by a planar point set. Graphs Comb. 18, 769\u2013779 (2002)","journal-title":"Graphs Comb."},{"key":"9783_CR17","doi-asserted-by":"crossref","unstructured":"Pach, J., Zeeuw, F. de.: Distinct distances on algebraic curves in the plane. Comb. Probab. Comput. (to appear)","DOI":"10.1145\/2582112.2582135"},{"key":"9783_CR18","doi-asserted-by":"crossref","first-page":"1484","DOI":"10.1016\/j.disc.2015.02.009","volume":"338","author":"OE Raz","year":"2015","unstructured":"Raz, O.E., Roche-Newton, O., Sharir, M.: Sets with few distinct distances do not have heavy lines. Discrete Math. 338, 1484\u20131492 (2015)","journal-title":"Discrete Math."},{"key":"9783_CR19","doi-asserted-by":"crossref","unstructured":"Raz, O.E., Sharir, M., Solymosi, J.: Polynomials vanishing on grids: the Elekes-R\u00f3nyai problem revisited. Am. J. Math. (to appear)","DOI":"10.1145\/2582112.2582150"},{"key":"9783_CR20","unstructured":"Raz, O.E., Sharir, M., de Zeeuw, F.: Polynomials vanishing on Cartesian products: the Elekes-Szab\u00f3 theorem revisited. Duke Math. J. (to appear)"},{"key":"9783_CR21","doi-asserted-by":"crossref","unstructured":"Sheffer, A.: Few distinct distances implies many points on a line. Blog post (2014)","DOI":"10.1007\/s00493-014-3180-6"},{"key":"9783_CR22","doi-asserted-by":"crossref","unstructured":"Sheffer, A., Zahl, J., de Zeeuw, F.: Few distinct distances implies no heavy lines or circles. Combinatorica. (to appear)","DOI":"10.1007\/s00493-014-3180-6"},{"key":"9783_CR23","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1007\/s00454-013-9526-9","volume":"50","author":"J Solymosi","year":"2013","unstructured":"Solymosi, J., Stojakovi\u0107, M.: Many collinear $$k$$ k -tuples with no $$k+ 1$$ k + 1 collinear points. Discrete Comput. Geom. 50, 811\u2013820 (2013)","journal-title":"Discrete Comput. Geom."},{"key":"9783_CR24","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/s00454-012-9420-x","volume":"48","author":"J Solymosi","year":"2012","unstructured":"Solymosi, J., Tao, T.: An incidence theorem in higher dimensions. Discrete Comput. Geom. 48, 255\u2013280 (2012)","journal-title":"Discrete Comput. Geom."},{"key":"9783_CR25","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1007\/BF02579194","volume":"3","author":"E Szemer\u00e9di","year":"1983","unstructured":"Szemer\u00e9di, E., Trotter, W.: Extremal problems in discrete geometry. Combinatorica 3, 381\u2013392 (1983)","journal-title":"Combinatorica"},{"key":"9783_CR26","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/S0001-8708(03)00004-5","volume":"180","author":"G Tardos","year":"2003","unstructured":"Tardos, G.: On distinct sums and distinct distances. Adv. Math. 180, 275\u2013289 (2003)","journal-title":"Adv. Math."},{"key":"9783_CR27","doi-asserted-by":"crossref","first-page":"545","DOI":"10.2307\/1969908","volume":"66","author":"H Whitney","year":"1957","unstructured":"Whitney, H.: Elementary structure of real algebraic varieties. Ann. Math. 66, 545\u2013556 (1957)","journal-title":"Ann. Math."}],"container-title":["Discrete & Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-016-9783-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00454-016-9783-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-016-9783-5","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,9]],"date-time":"2019-09-09T10:35:53Z","timestamp":1568025353000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00454-016-9783-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,6,8]]},"references-count":27,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2016,9]]}},"alternative-id":["9783"],"URL":"https:\/\/doi.org\/10.1007\/s00454-016-9783-5","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,6,8]]}}}