{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,1]],"date-time":"2025-05-01T00:35:39Z","timestamp":1746059739463},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,10,12]],"date-time":"2023-10-12T00:00:00Z","timestamp":1697068800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,10,12]],"date-time":"2023-10-12T00:00:00Z","timestamp":1697068800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2024,2]]},"abstract":"Abstract<\/jats:title>We show that if a finite point set $$P\\subseteq {\\mathbb {R}}^2$$<\/jats:tex-math>\n \n P<\/mml:mi>\n \u2286<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> has the fewest congruence classes of triangles possible, up to a constant M<\/jats:italic>, then at least one of the following holds.\n \n There is a $$\\sigma >0$$<\/jats:tex-math>\n \n \u03c3<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and a line l<\/jats:italic> which contains $$\\Omega (|P|^\\sigma )$$<\/jats:tex-math>\n \n \u03a9<\/mml:mi>\n (<\/mml:mo>\n |<\/mml:mo>\n P<\/mml:mi>\n \n |<\/mml:mo>\n \u03c3<\/mml:mi>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> points of P<\/jats:italic>. Further, a positive proportion of P<\/jats:italic> is covered by lines parallel to l<\/jats:italic> each containing $$\\Omega (|P|^\\sigma )$$<\/jats:tex-math>\n \n \u03a9<\/mml:mi>\n (<\/mml:mo>\n |<\/mml:mo>\n P<\/mml:mi>\n \n |<\/mml:mo>\n \u03c3<\/mml:mi>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> points of P<\/jats:italic>.<\/jats:p>\n <\/jats:list-item>\n \n There is a circle $$\\gamma $$<\/jats:tex-math>\n \u03b3<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> which contains a positive proportion of P<\/jats:italic>.<\/jats:p>\n <\/jats:list-item>\n <\/jats:list> This provides evidence for two conjectures of Erd\u0151s. We use the result of Petridis\u2013Roche\u2013Newton\u2013Rudnev\u2013Warren on the structure of the affine group combined with classical results from additive combinatorics.<\/jats:p>","DOI":"10.1007\/s00493-023-00066-z","type":"journal-article","created":{"date-parts":[[2023,10,12]],"date-time":"2023-10-12T11:03:14Z","timestamp":1697108594000},"page":"155-178","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Structural Theorem for Sets with Few Triangles"],"prefix":"10.1007","volume":"44","author":[{"given":"Sam","family":"Mansfield","sequence":"first","affiliation":[]},{"given":"Jonathan","family":"Passant","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,10,12]]},"reference":[{"key":"66_CR1","first-page":"10","volume":"43","author":"A Balog","year":"2007","unstructured":"Balog, A.: Many additive quadruples. CRM Proc. Lecture Notes Addit. Comb. 43, 10 (2007)","journal-title":"CRM Proc. Lecture Notes Addit. Comb."},{"issue":"3","key":"66_CR2","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1007\/BF01212974","volume":"14","author":"A Balog","year":"1994","unstructured":"Balog, A., Szemer\u00e9di, E.: A statistical theorem of set addition. Combinatorica 14(3), 263\u2013268 (1994)","journal-title":"Combinatorica"},{"issue":"4","key":"66_CR3","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1007\/BF01194999","volume":"17","author":"G Elekes","year":"1997","unstructured":"Elekes, G.: On linear combinatorics I. Concurrency-an algebraic approach. Combinatorica 17(4), 447\u2013458 (1997)","journal-title":"Combinatorica"},{"issue":"4","key":"66_CR4","doi-asserted-by":"publisher","first-page":"365","DOI":"10.4064\/aa-81-4-365-367","volume":"81","author":"G Elekes","year":"1997","unstructured":"Elekes, G.: On the number of sums and products. Acta Arith. 81(4), 365\u2013367 (1997)","journal-title":"Acta Arith."},{"issue":"1","key":"66_CR5","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1007\/PL00009806","volume":"18","author":"Gy\u00f6rgy Elekes","year":"1998","unstructured":"Elekes, Gy\u00f6rgy.: On linear combinatorics II. Structure theorems via additive number theory. Combinatorica 18(1), 13\u201325 (1998)","journal-title":"Combinatorica"},{"issue":"11","key":"66_CR6","first-page":"241","volume":"II","author":"G Elekes","year":"2001","unstructured":"Elekes, G.: Sums versus products in number theory, algebra and Erd\u0151s geometry. Paul Erd\u0151s Math. II(11), 241\u2013290 (2001)","journal-title":"Paul Erd\u0151s Math."},{"issue":"4","key":"66_CR7","doi-asserted-by":"publisher","first-page":"571","DOI":"10.1017\/S0963548311000137","volume":"20","author":"G Elekes","year":"2011","unstructured":"Elekes, G., Sharir, M.: Incidences in three dimensions and distinct distances in the plane. Comb. Probab. Comput. 20(4), 571\u2013608 (2011)","journal-title":"Comb. Probab. Comput."},{"key":"66_CR8","doi-asserted-by":"publisher","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. Discret. Math. 60, 147\u2013153 (1986)","journal-title":"Discret. Math."},{"issue":"1\u20133","key":"66_CR9","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1016\/0012-365X(93)90155-M","volume":"111","author":"P Erd\u0151s","year":"1993","unstructured":"Erd\u0151s, P., F\u00fcredi, Z., Pach, J., Ruzsa, I.Z.: The grid revisited. Discret. Math. 111(1\u20133), 189\u2013196 (1993)","journal-title":"Discret. Math."},{"issue":"7","key":"66_CR10","doi-asserted-by":"publisher","first-page":"738","DOI":"10.1080\/00029890.1970.11992573","volume":"77","author":"P Erd\u0151s","year":"1946","unstructured":"Erd\u0151s, P.: On sets of distances of n points. Am. Math. Mon. 77(7), 738\u2013740 (1946)","journal-title":"Am. Math. Mon."},{"issue":"103","key":"66_CR11","first-page":"99","volume":"4","author":"P Erd\u0151s","year":"1975","unstructured":"Erd\u0151s, P.: On some problems of elementary and combinatorial geometry. Ann. Math. Ser. 4(103), 99\u2013108 (1975)","journal-title":"Ann. Math. Ser."},{"issue":"3","key":"66_CR12","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1007\/s00039-001-0332-9","volume":"11","author":"WT Gowers","year":"2001","unstructured":"Gowers, W.T.: A new proof of Szemer\u00e9di\u2019s theorem. Geom. Funct. Anal. GAFA 11(3), 465\u2013588 (2001)","journal-title":"Geom. Funct. Anal. GAFA"},{"issue":"1","key":"66_CR13","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1112\/jlms\/jdl021","volume":"75","author":"B Green","year":"2007","unstructured":"Green, B., Ruzsa, I.Z.: Freiman\u2019s theorem in an arbitrary Abelian group. J. Lond. Math. Soc. 75(1), 163\u2013175 (2007)","journal-title":"J. Lond. Math. Soc."},{"key":"66_CR14","doi-asserted-by":"crossref","unstructured":"Guth, L., Katz, N.H.: On the Erd\u0151s distinct distances problem in the plane. Ann. Math. 155\u2013190 (2015)","DOI":"10.4007\/annals.2015.181.1.2"},{"issue":"5","key":"66_CR15","doi-asserted-by":"publisher","first-page":"1095","DOI":"10.1007\/s00493-016-3665-6","volume":"38","author":"B Hanson","year":"2018","unstructured":"Hanson, B.: The additive structure of Cartesian products spanning few distinct distances. Combinatorica 38(5), 1095\u20131100 (2018)","journal-title":"Combinatorica"},{"key":"66_CR16","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1016\/j.aim.2014.11.014","volume":"271","author":"J Koll\u00e1r","year":"2015","unstructured":"Koll\u00e1r, J.: Szemer\u00e9di-Trotter-type theorems in dimension 3. Adv. Math. 271, 30\u201361 (2015)","journal-title":"Adv. Math."},{"key":"66_CR17","doi-asserted-by":"crossref","unstructured":"K\u0151v\u00e1ri, T., S\u00f3s, V.T., Tur\u00e1n, P.: On a problem of Zarankiewicz. In: Colloquium Mathematicum, vol.\u00a03, pp. 50\u201357. Polska Akademia Nauk (1954)","DOI":"10.4064\/cm-3-1-50-57"},{"key":"66_CR18","unstructured":"Landau, E.: \u00dcber die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate (1909)"},{"issue":"2","key":"66_CR19","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/s00454-016-9783-5","volume":"56","author":"B Lund","year":"2016","unstructured":"Lund, B., Sheffer, A., De Zeeuw, F.: Bisector energy and few distinct distances. Discret. Comput. Geom. 56(2), 337\u2013356 (2016)","journal-title":"Discret. Comput. Geom."},{"issue":"3","key":"66_CR20","doi-asserted-by":"publisher","first-page":"965","DOI":"10.1093\/qmath\/haz006","volume":"70","author":"A Mudgal","year":"2019","unstructured":"Mudgal, A.: Sums of linear transformations in higher dimensions. Q. J. Math. 70(3), 965\u2013984 (2019)","journal-title":"Q. J. Math."},{"issue":"1","key":"66_CR21","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1017\/S0963548316000225","volume":"26","author":"J Pach","year":"2017","unstructured":"Pach, J., De Zeeuw, F.: Distinct distances on algebraic curves in the plane. Comb. Probab. Comput. 26(1), 99\u2013117 (2017)","journal-title":"Comb. Probab. Comput."},{"issue":"2","key":"66_CR22","doi-asserted-by":"publisher","first-page":"1154","DOI":"10.1093\/imrn\/rnaa130","volume":"2022","author":"G Petridis","year":"2022","unstructured":"Petridis, G., Roche-Newton, O., Rudnev, M., Warren, A.: An energy bound in the affine group. Int. Math. Res. Not. 2022(2), 1154\u20131172 (2022)","journal-title":"Int. Math. Res. Not."},{"issue":"1","key":"66_CR23","first-page":"P1","volume":"26","author":"C Pohoata","year":"2019","unstructured":"Pohoata, C.: On Cartesian products which determine few distinct distances. Electron. J. Comb. 26(1), P1 (2019)","journal-title":"Electron. J. Comb."},{"issue":"8","key":"66_CR24","doi-asserted-by":"publisher","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. Discret. Math. 338(8), 1484\u20131492 (2015)","journal-title":"Discret. Math."},{"key":"66_CR25","unstructured":"Roche-Newton, O.: On sets with few distinct distances. arXiv preprint arXiv:1608.02775 (2016)"},{"key":"66_CR26","unstructured":"Rudnev, M.: On the number of classes of triangles determined by $$ n $$ points in $$\\mathbb{R}^{2}$$. arXiv preprint arXiv:1205.4865 (2012)"},{"issue":"1","key":"66_CR27","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1007\/s00493-016-3329-6","volume":"38","author":"M Rudnev","year":"2018","unstructured":"Rudnev, M.: On the number of incidences between points and planes in three dimensions. Combinatorica 38(1), 219\u2013254 (2018)","journal-title":"Combinatorica"},{"issue":"3","key":"66_CR28","doi-asserted-by":"publisher","first-page":"738","DOI":"10.1112\/mtk.12120","volume":"68","author":"M Rudnev","year":"2022","unstructured":"Rudnev, M., Shkredov, I.D.: On the growth rate in SL$$_2(\\mathbb{F} _p)$$, the affine group and sum-product type implications. Mathematika 68(3), 738\u2013783 (2022)","journal-title":"Mathematika"},{"key":"66_CR29","unstructured":"Sheffer, A.: Distinct distances: open problems and current bounds. arXiv preprint arXiv:1406.1949 (2014)"},{"key":"66_CR30","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"},{"issue":"3","key":"66_CR31","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1007\/s00493-014-3180-6","volume":"36","author":"A Sheffer","year":"2016","unstructured":"Sheffer, A., Zahl, J., De Zeeuw, F.: Few distinct distances implies no heavy lines or circles. Combinatorica 36(3), 349\u2013364 (2016)","journal-title":"Combinatorica"},{"key":"66_CR32","doi-asserted-by":"publisher","first-page":"182","DOI":"10.1016\/j.jnt.2020.06.014","volume":"220","author":"ID Shkredov","year":"2021","unstructured":"Shkredov, I.D.: Modular hyperbolas and bilinear forms of Kloosterman sums. J. Number Theory 220, 182\u2013211 (2021)","journal-title":"J. Number Theory"},{"issue":"1\u20132","key":"66_CR33","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1016\/S0012-365X(01)00441-1","volume":"256","author":"YV Stanchescu","year":"2002","unstructured":"Stanchescu, Y.V.: Planar sets containing no three collinear points and non-averaging sets of integers. Discret. Math. 256(1\u20132), 387\u2013395 (2002)","journal-title":"Discret. Math."}],"container-title":["Combinatorica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-023-00066-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00493-023-00066-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-023-00066-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,21]],"date-time":"2024-02-21T22:03:30Z","timestamp":1708553010000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00493-023-00066-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,12]]},"references-count":33,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,2]]}},"alternative-id":["66"],"URL":"https:\/\/doi.org\/10.1007\/s00493-023-00066-z","relation":{},"ISSN":["0209-9683","1439-6912"],"issn-type":[{"value":"0209-9683","type":"print"},{"value":"1439-6912","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,12]]},"assertion":[{"value":"9 September 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 June 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 August 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 October 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}