{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T12:07:53Z","timestamp":1777464473046,"version":"3.51.4"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2015,9,3]],"date-time":"2015-09-03T00:00:00Z","timestamp":1441238400000},"content-version":"tdm","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":[[2015,12]]},"DOI":"10.1007\/s00454-015-9731-9","type":"journal-article","created":{"date-parts":[[2015,9,2]],"date-time":"2015-09-02T14:59:00Z","timestamp":1441205940000},"page":"759-770","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["New Results for the Growth of Sets of Real Numbers"],"prefix":"10.1007","volume":"54","author":[{"given":"Timothy G. F.","family":"Jones","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2015,9,3]]},"reference":[{"issue":"3","key":"9731_CR1","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1137\/050641375","volume":"21","author":"R Apfelbaum","year":"2007","unstructured":"Apfelbaum, R., Sharir, M.: Large complete bipartite subgraphs in incidence graphs of points and hyperplanes. SIAM J. Discrete Math. 21(3), 707\u2013725 (2007)","journal-title":"SIAM J. Discrete Math."},{"issue":"1","key":"9731_CR2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1142\/S1793042105000108","volume":"1","author":"J Bourgain","year":"2005","unstructured":"Bourgain, J.: More on the sum-product phenomenon in prime fields and its applications. Int. J. Number Theory 1(1), 1\u201332 (2005)","journal-title":"Int. J. Number Theory"},{"issue":"1","key":"9731_CR3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0305004108001230","volume":"146","author":"J Bourgain","year":"2009","unstructured":"Bourgain, J., Garaev, M.: On a variant of sum-product estimates and explicit exponential sum bounds in prime fields. Math. Proc. Camb. Philos. Soc. 146(1), 1\u201321 (2009)","journal-title":"Math. Proc. Camb. Philos. Soc."},{"issue":"1","key":"9731_CR4","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s00039-004-0451-1","volume":"14","author":"J Bourgain","year":"2004","unstructured":"Bourgain, J., Katz, N., Tao, T.: A sum-product estimate in finite fields and applications. Geom. Funct. Anal. 14(1), 27\u201357 (2004)","journal-title":"Geom. Funct. Anal."},{"issue":"2","key":"9731_CR5","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1007\/BF02187785","volume":"5","author":"H Edelsbrunner","year":"1990","unstructured":"Edelsbrunner, H., Guibas, L., Sharir, M.: The complexity of many cells in arrangements of planes and related problems. Discrete Comput. Geom. 5(2), 197\u2013216 (1990)","journal-title":"Discrete Comput. Geom."},{"issue":"4","key":"9731_CR6","doi-asserted-by":"crossref","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":"9731_CR7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/jcta.1999.2976","volume":"89","author":"G Elekes","year":"2000","unstructured":"Elekes, G., Ronyai, G.: A combinatorial problem on polynomials and rational functions. J. Comb. Theory, Ser. A 89(1), 1\u201320 (2000)","journal-title":"J. Comb. Theory, Ser. A"},{"key":"9731_CR8","doi-asserted-by":"crossref","unstructured":"Elekes, G., Sharir, M.: Incidences in three dimensions and distinct distances in the plane. In: Proceedings 26th ACM Symposium on Computational Geometry, pp. 413\u2013422 (2010)","DOI":"10.1145\/1810959.1811028"},{"issue":"1\u20132","key":"9731_CR9","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1023\/A:1009709908223","volume":"2","author":"K Ford","year":"1998","unstructured":"Ford, K.: Sums and products from a finite set of real numbers. Ramanujan J. 2(1\u20132), 59\u201366 (1998)","journal-title":"Ramanujan J."},{"key":"9731_CR10","first-page":"1","volume":"11","author":"M Garaev","year":"2007","unstructured":"Garaev, M.: An explicit sum-product estimate in $$\\mathbb{F}_p$$ F p . Int. Math. Res. Not. 11, 1\u201311 (2007)","journal-title":"Int. Math. Res. Not."},{"issue":"8","key":"9731_CR11","doi-asserted-by":"crossref","first-page":"2735","DOI":"10.1090\/S0002-9939-08-09386-6","volume":"136","author":"M Garaev","year":"2008","unstructured":"Garaev, M.: The sum-product estimate for large subsets of prime fields. Proc. Am. Math. Soc. 136(8), 2735\u20132739 (2008)","journal-title":"Proc. Am. Math. Soc."},{"issue":"1","key":"9731_CR12","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s00209-009-0504-0","volume":"265","author":"M Garaev","year":"2010","unstructured":"Garaev, M., Shen, C.: On the size of the set $${A(A+1)}$$ A ( A + 1 ) . Math. Z. 265(1), 125\u2013132 (2010)","journal-title":"Math. Z."},{"issue":"1","key":"9731_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.: On the Erd\u00f6s distinct distance problem in the plane. Ann. Math. 181(1), 155\u2013190 (2015)","journal-title":"Ann. Math."},{"issue":"4","key":"9731_CR14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4310\/MRL.2011.v18.n4.a8","volume":"18","author":"A Iosevich","year":"2011","unstructured":"Iosevich, A., Roche-Newton, O., Rudnev, M.: On an application of Guth\u2013Katz theorem. Math. Res. Lett. 18(4), 1\u20137 (2011)","journal-title":"Math. Res. Lett."},{"issue":"3","key":"9731_CR15","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1016\/j.jcta.2012.11.001","volume":"120","author":"T Jones","year":"2013","unstructured":"Jones, T., Roche-Newton, O.: Improved bounds on the set A(A+1). J. Comb. Theory, Ser. A. 120(3), 515\u2013526 (2013)","journal-title":"J. Comb. Theory, Ser. A."},{"key":"9731_CR16","doi-asserted-by":"crossref","unstructured":"Katz, N.H., Shen, C.-Y: Garaev\u2019s inequality in finite fields not of prime order. Online J. Anal. Comb. (3), Article #3, 1\u20136 (2008)","DOI":"10.61091\/ojac-303"},{"issue":"7","key":"9731_CR17","doi-asserted-by":"crossref","first-page":"2499","DOI":"10.1090\/S0002-9939-08-09385-4","volume":"136","author":"N Katz","year":"2008","unstructured":"Katz, N., Shen, C.: A slight improvement to Garaev\u2019s sum product estimate. Proc. Am. Math. Soc. 136(7), 2499\u20132504 (2008)","journal-title":"Proc. Am. Math. Soc."},{"key":"9731_CR18","unstructured":"Konyagin, S., Rudnev, M.: On new sum-product type estimates, Preprint (2012). arXiv:1207.6785"},{"issue":"2","key":"9731_CR19","doi-asserted-by":"crossref","first-page":"153","DOI":"10.4064\/aa147-2-4","volume":"147","author":"L Li","year":"2011","unstructured":"Li, L.: Slightly improved sum-product estimates in fields of prime order. Acta Arith. 147(2), 153\u2013160 (2011)","journal-title":"Acta Arith."},{"key":"9731_CR20","unstructured":"Li, L., Roche-Newton, O.: Convexity and a sum-product type estimate, Preprint (2011). arXiv:1111.5159"},{"issue":"3","key":"9731_CR21","doi-asserted-by":"crossref","first-page":"1285","DOI":"10.1137\/110823122","volume":"25","author":"L Li","year":"2011","unstructured":"Li, L., Roche-Newton, O.: An improved sum-product estimate for general finite fields. SIAM J. Discrete Math. 25(3), 1285\u20131296 (2011)","journal-title":"SIAM J. Discrete Math."},{"issue":"1","key":"9731_CR22","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1090\/S0002-9939-97-03510-7","volume":"125","author":"M Nathanson","year":"1997","unstructured":"Nathanson, M.: On sums and products of integers. Proc. Am. Math. Soc. 125(1), 9\u201316 (1997)","journal-title":"Proc. Am. Math. Soc."},{"key":"9731_CR23","doi-asserted-by":"crossref","unstructured":"Raz, O., Sharir, M., Solymosi, J.: Polynomials vanishing on grids: the Elekes\u2013Rnyai problem revisited. In: Proceedings of the 30th ACM Symposium on Computational Geometry, pp. 251\u2013260 (2014)","DOI":"10.1145\/2582112.2582150"},{"key":"9731_CR24","doi-asserted-by":"crossref","first-page":"3693","DOI":"10.1093\/imrn\/rnr158","volume":"16","author":"M Rudnev","year":"2012","unstructured":"Rudnev, M.: An improved sum-product inequality in fields of prime order. Int. Math. Res. Not. 16, 3693\u20133704 (2012)","journal-title":"Int. Math. Res. Not."},{"key":"9731_CR25","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-3896-6","volume-title":"Projective Geometry","author":"P Samuel","year":"1988","unstructured":"Samuel, P.: Projective Geometry. Springer-Verlag, Berlin (1988)"},{"issue":"1","key":"9731_CR26","doi-asserted-by":"crossref","first-page":"N40","DOI":"10.37236\/915","volume":"15","author":"C Shen","year":"2008","unstructured":"Shen, C.: Quantitative sum product estimates on different sets. Electron. J. Comb. 15(1), N40 (2008)","journal-title":"Electron. J. Comb."},{"issue":"3","key":"9731_CR27","doi-asserted-by":"crossref","first-page":"921","DOI":"10.5802\/jtnb.527","volume":"17","author":"J Solymosi","year":"2005","unstructured":"Solymosi, J.: On sum-sets and product-sets of complex numbers. J. Th\u00e9or. Nombres Bordx. 17(3), 921\u2013924 (2005)","journal-title":"J. Th\u00e9or. Nombres Bordx."},{"issue":"4","key":"9731_CR28","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1112\/S0024609305004261","volume":"37","author":"J Solymosi","year":"2005","unstructured":"Solymosi, J.: On the number of sums and products. Bull. Lond. Math. Soc. 37(4), 491\u2013494 (2005)","journal-title":"Bull. Lond. Math. Soc."},{"key":"9731_CR29","doi-asserted-by":"crossref","unstructured":"Szemer\u00e9di, E., Trotter Jr., W.T.: Extremal problems in discrete geometry. Combinatorica 3(3\u20134), 381\u2013392 (1983)","DOI":"10.1007\/BF02579194"}],"container-title":["Discrete & Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-015-9731-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00454-015-9731-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-015-9731-9","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,30]],"date-time":"2025-05-30T10:58:57Z","timestamp":1748602737000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00454-015-9731-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,9,3]]},"references-count":29,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2015,12]]}},"alternative-id":["9731"],"URL":"https:\/\/doi.org\/10.1007\/s00454-015-9731-9","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,9,3]]}}}