{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T04:55:31Z","timestamp":1768452931723,"version":"3.49.0"},"reference-count":39,"publisher":"Duke University Press","issue":"3-4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Notre Dame J. Formal Logic"],"published-print":{"date-parts":[[2013,1,1]]},"DOI":"10.1215\/00294527-2143952","type":"journal-article","created":{"date-parts":[[2013,8,9]],"date-time":"2013-08-09T13:09:43Z","timestamp":1376053783000},"source":"Crossref","is-referenced-by-count":13,"title":["On Pseudo-Finite Dimensions"],"prefix":"10.1215","volume":"54","author":[{"given":"Ehud","family":"Hrushovski","sequence":"first","affiliation":[]}],"member":"73","reference":[{"key":"13","unstructured":"[13] Goldbring, I., and H. Towsner, \u201cAn approximate logic for measures,\u201d preprint, <a href=\"arXiv:1106.2854v2\">arXiv:1106.2854v2<\/a> [math.LO]."},{"key":"29","unstructured":"[29] Pyber, L., and E. Szab\u00f3, \u201cGrowth in finite simple groups of Lie type of bounded rank,\u201d preprint, <a href=\"arXiv:1005.1858v2\">arXiv:1005.1858v2<\/a> [math.GR]."},{"key":"34","unstructured":"[34] Shelah, S., \u201cDefinable groups for dependent and $2$-dependent theories,\u201d preprint, <a href=\"arXiv:math\/0703045v4\">arXiv:math\/0703045v4<\/a> [math.LO]."},{"key":"35","unstructured":"[35] Schwartz, D., J. Solymosi, and F. de Zeeuw, \u201cExtensions of a result of Elekes and R\u00f3nyai,\u201d preprint, <a href=\"arXiv:1206.2717v1\">arXiv:1206.2717v1<\/a> [math.CO]."},{"key":"38","unstructured":"[38] Tao, T., \u201cExpanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets,\u201d preprint, <a href=\"arXiv:1211.2894v4\">arXiv:1211.2894v4<\/a> [math.CO]."},{"key":"39","doi-asserted-by":"crossref","unstructured":"[39] Yaacov, I. B., and A. Usvyatsov, \u201cContinuous first order logic and local stability,\u201d <i>Transactions of the American Mathematical Society<\/i>, vol. 362 (2010), pp. 5213\u201359, <a href=\"arXiv:0801.4303\">arXiv:0801.4303<\/a>.","DOI":"10.1090\/S0002-9947-10-04837-3"},{"key":"1","doi-asserted-by":"crossref","unstructured":"[1] Bourgain, J., N. H. Katz, and T. Tao, \u201cA sum-product estimate in finite fields, and applications,\u201d <i>Geometric and Functional Analysis<\/i>, vol. 14 (2004), pp. 27\u201357.","DOI":"10.1007\/s00039-004-0451-1"},{"key":"2","doi-asserted-by":"crossref","unstructured":"[2] Breuillard, E., Y. Cornulier, A. Lubotzky, and C. Meiri, \u201cOn conjugacy growth of linear groups,\u201d <i>Mathematical Proceedings of the Cambridge Philosophical Society<\/i>, vol. 154 (2013), pp. 261\u201377.","DOI":"10.1017\/S030500411200059X"},{"key":"3","doi-asserted-by":"crossref","unstructured":"[3] Breuillard, E., B. Green, and T. Tao, \u201cApproximate subgroups of linear groups,\u201d <i>Geometric and Functional Analysis<\/i>, vol. 21 (2011), pp. 774\u2013819.","DOI":"10.1007\/s00039-011-0122-y"},{"key":"4","doi-asserted-by":"crossref","unstructured":"[4] Breuillard, E., B. Green, and T. Tao, \u201cLinear approximate groups,\u201d <i>Electronic Research Announcements in Mathematical Sciences<\/i>, vol. 17(2010), pp. 57\u201367.","DOI":"10.3934\/era.2010.17.57"},{"key":"5","doi-asserted-by":"crossref","unstructured":"[5] Bukh, B., and J. Tsimerman, \u201cSum-product estimates for rational functions,\u201d <i>Proceedings of the London Mathematical Society (3)<\/i>, vol. 104 (2012), pp. 1\u201326.","DOI":"10.1112\/plms\/pdr018"},{"key":"6","doi-asserted-by":"crossref","unstructured":"[6] Blossier, T., A. Martin-Pizarro, and F. O. Wagner, \u201cOn variants of CM-triviality,\u201d <i>Fundamenta Mathematicae<\/i>, vol. 219 (2012), pp. 253\u201362.","DOI":"10.4064\/fm219-3-4"},{"key":"7","doi-asserted-by":"crossref","unstructured":"[7] Cherlin, G., \u201cTwo problems on homogeneous structures, revisited,\u201d pp. 319\u2013416 in <i>Model Theoretic Methods in Finite Combinatorics<\/i>, edited by M. Grohe and J. A. Makowsky, vol. 558 of <i>Contemporary Mathematics<\/i>, American Mathematical Society, Providence, 2011.","DOI":"10.1090\/conm\/558\/11055"},{"key":"8","doi-asserted-by":"crossref","unstructured":"[8] Cherlin, G., L. Harrington, and A. H. Lachlan, \u201c$\\aleph_{0}$-categorical, $\\aleph_{0}$-stable structures,\u201d <i>Annals of Pure and Applied Logic<\/i>, vol. 28 (1985), pp. 103\u201335.","DOI":"10.1016\/0168-0072(85)90023-5"},{"key":"9","unstructured":"[9] Cherlin, G., and E. Hrushovski, <i>Finite Structures with Few Types<\/i>, vol. 152 of <i>Annals of Mathematics Studies<\/i>, Princeton University Press, Princeton, 2003."},{"key":"10","unstructured":"[10] Elekes, G., \u201cSUMS versus PRODUCTS in number theory, algebra and Erd\u0151s geometry,\u201d pp. 241\u201390 in <i>Paul Erd\u0151s and his Mathematics, II (Budapest, 1999)<\/i>, vol. 11 of <i>Bolyai Society Mathematical Studies<\/i>, J\u00e1nos Bolyai Mathematical Society, Budapest, 2002."},{"key":"11","doi-asserted-by":"crossref","unstructured":"[11] Elekes, G., and E. Szab\u00f3, \u201cHow to find groups? (and how to use them in Erd\u0151s geometry?),\u201d <i>Combinatorica<\/i>, vol. 32 (2012), pp. 537\u201371.","DOI":"10.1007\/s00493-012-2505-6"},{"key":"12","doi-asserted-by":"crossref","unstructured":"[12] Erd\u0151s, P., and E. Szemer\u00e9di, \u201cOn sums and products of integers,\u201d pp. 213\u201318 in <i>Studies in Pure Mathematics<\/i>, Birkh\u00e4user, Basel, 1983.","DOI":"10.1007\/978-3-0348-5438-2_19"},{"key":"14","doi-asserted-by":"crossref","unstructured":"[14] Green, B., \u201cFinite field models in additive combinatorics,\u201d pp. 1\u201327 in <i>Surveys in Combinatorics 2005<\/i>, vol. 327 of <i>London Mathematical Society Lecture Notes Series<\/i>, Cambridge University Press, Cambridge, 2005.","DOI":"10.1017\/CBO9780511734885.002"},{"key":"15","doi-asserted-by":"crossref","unstructured":"[15] Hrushovski, E., \u201cUnimodular minimal structures,\u201d <i>Journal of the London Mathematical Society (2)<\/i>, vol. 46 (1992), pp. 385\u201396.","DOI":"10.1112\/jlms\/s2-46.3.385"},{"key":"16","doi-asserted-by":"crossref","unstructured":"[16] Hrushovski, E., \u201cStable group theory and approximate subgroups,\u201d <i>Journal of the American Mathematical Society<\/i>, vol. 25 (2012), pp. 189\u2013243.","DOI":"10.1090\/S0894-0347-2011-00708-X"},{"key":"17","doi-asserted-by":"crossref","unstructured":"[17] Hrushovski, E. [U. Hrushovski], and A. Pillay, \u201cWeakly normal groups,\u201d pp. 233\u201344 in <i>Logic Colloquium \u201985 (Orsay, 1985)<\/i>, vol. 122of <i>Studies in Logic and the Foundations of Mathematics<\/i>, North-Holland, Amsterdam, 1987.","DOI":"10.1016\/S0049-237X(09)70556-7"},{"key":"18","doi-asserted-by":"crossref","unstructured":"[18] Hrushovski, E., and A. Pillay, \u201cGroups definable in local fields and pseudo-finite fields,\u201d <i>Israel Journal of Mathematics<\/i>, vol. 85 (1994), pp. 203\u201362.","DOI":"10.1007\/BF02758643"},{"key":"19","doi-asserted-by":"crossref","unstructured":"[19] Hrushovski, E., and F. Wagner, \u201cCounting and dimensions,\u201d pp. 161\u201376 in <i>Model Theory with Applications to Algebra and Analysis, Vol. 2<\/i>, vol. 350 of <i>London Mathematical Society Lecture Note Series<\/i>, Cambridge University Press, Cambridge, 2008.","DOI":"10.1017\/CBO9780511735219.005"},{"key":"20","doi-asserted-by":"crossref","unstructured":"[20] Jordan, C., \u201cM\u00e9moire sur les \u00e9quations diff\u00e9rentielles lin\u00e9aires \u00e0 int\u00e9grale alg\u00e9brique,\u201d <i>Crelle<\/i>, vol. 84 (1878), pp. 89\u2013215.","DOI":"10.1515\/9783112335789-007"},{"key":"21","doi-asserted-by":"publisher","unstructured":"[21] Kantor, W. M., M. W. Liebeck, and H. D. Macpherson, \u201c$\\aleph_{0}$-categorical structures smoothly approximated by finite substructures,\u201d <i>Proceedings of the London Mathematical Society (3)<\/i>, vol. 59 (1989), pp. 439\u201363.","DOI":"10.1112\/plms\/s3-59.3.439"},{"key":"22","doi-asserted-by":"crossref","unstructured":"[22] Keisler, H. J., \u201cProbability quantifiers,\u201d pp. 509\u201356 in <i>Model-Theoretic Logics<\/i>, edited by J. Barwise and S. Feferman, Springer, New York, 1985.","DOI":"10.1017\/9781316717158.021"},{"key":"23","doi-asserted-by":"crossref","unstructured":"[23] Kim, B., and A. Pillay, \u201cSimple theories,\u201d pp. 149\u201364 in <i>Joint AILA-KGS Model Theory Meeting (Florence, 1995)<\/i>, vol. 88 of <i>Annals of Pure and Applied Logic<\/i>, Elsevier, Amsterdam, 1997.","DOI":"10.1016\/S0168-0072(97)00019-5"},{"key":"24","doi-asserted-by":"crossref","unstructured":"[24] Larsen, M. J., and R. Pink, \u201cFinite subgroups of algebraic groups,\u201d <i>Journal of the American Mathematical Society<\/i>, vol. 24 (2011), 1105\u201358.","DOI":"10.1090\/S0894-0347-2011-00695-4"},{"key":"25","doi-asserted-by":"crossref","unstructured":"[25] Macpherson, D., and C. Steinhorn, \u201cOne-dimensional asymptotic classes of finite structures,\u201d <i>Transactions of the American Mathematical Society<\/i>, vol. 360 (2008), pp. 411\u201348.","DOI":"10.1090\/S0002-9947-07-04382-6"},{"key":"26","doi-asserted-by":"crossref","unstructured":"[26] Martin, G. A., \u201cDefinability in reducts of algebraically closed fields,\u201d <i>Journal of Symbolic Logic<\/i>, vol. 53 (1988), pp. 188\u201399.","DOI":"10.1017\/S0022481200029029"},{"key":"27","doi-asserted-by":"crossref","unstructured":"[27] Nikolov, N., and L. Pyber, \u201cProduct decompositions of quasirandom groups and a Jordan type theorem,\u201d <i>Journal of the European Mathematical Society (JEMS)<\/i>, vol. 13 (2011), 1063\u201377.","DOI":"10.4171\/JEMS\/275"},{"key":"28","unstructured":"[28] Pillay, A., \u201cForking in the category of existentially closed structures,\u201d pp. 23\u201342 in <i>Connections Between Model Theory and Algebraic and Analytic Geometry<\/i>, vol. 6 of <i>Quaderni di Matematica<\/i>, Department of Mathematics Seconda University of Napoli, Caserta, 2000."},{"key":"30","unstructured":"[30] Rabinovich, E. D., <i>Definability of a Field in Sufficiently Rich Incidence Systems<\/i>, with an introduction by Wilfrid Hodges, vol. 14 of <i>QMW Maths Notes<\/i>, Queen Mary and Westfield College, School of Mathematical Sciences, London, 1993."},{"key":"31","doi-asserted-by":"crossref","unstructured":"[31] Ramsey, F. P., \u201cOn a problem of formal logic,\u201d <i>Proceedings of the London Mathematical Society<\/i>, vol. S2-30 (1930), pp. 264\u201385.","DOI":"10.1112\/plms\/s2-30.1.264"},{"key":"32","doi-asserted-by":"crossref","unstructured":"[32] Sanders, T., \u201cOn a nonabelian Balog\u2013Szemer\u00e9di-type lemma,\u201d <i>Journal of the Australian Mathematical Society<\/i>, vol. 89 (2010), pp. 127\u201332.","DOI":"10.1017\/S1446788710000236"},{"key":"33","unstructured":"[33] Shelah, S., \u201cThe lazy model-theoretician\u2019s guide to stability,\u201d pp. 241\u2013308 in <i>Comptes Rendus de la Semaine d\u2019\u00c9tude en Th\u00e9orie des Mod\u00e8les (Louvain-la-Neuve, 1975)<\/i>, vol. 18 of <i>Logique et Analyse (N.S.)<\/i>, Centre National Belge Recherches de Logique, Brussels, 1975."},{"key":"36","doi-asserted-by":"crossref","unstructured":"[36] Solymosi, J., and T. Tao, \u201cAn incidence theorem in higher dimensions,\u201d <i>Discrete and Computational Geometry<\/i>, vol. 48 (2012), pp. 255\u201380.","DOI":"10.1007\/s00454-012-9420-x"},{"key":"37","unstructured":"[37] Tao, T., \u201cThe sum-product phenomenon in arbitrary rings,\u201d <i>Contributions to Discrete Mathematics<\/i>, vol. 4 (2009), pp. 59\u201382."}],"container-title":["Notre Dame Journal of Formal Logic"],"original-title":[],"link":[{"URL":"https:\/\/projecteuclid.org\/journalArticle\/Download?urlid=10.1215\/00294527-2143952","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,30]],"date-time":"2024-01-30T20:58:10Z","timestamp":1706648290000},"score":1,"resource":{"primary":{"URL":"https:\/\/projecteuclid.org\/journals\/notre-dame-journal-of-formal-logic\/volume-54\/issue-3-4\/On-Pseudo-Finite-Dimensions\/10.1215\/00294527-2143952.full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,1,1]]},"references-count":39,"journal-issue":{"issue":"3-4","published-online":{"date-parts":[[2013,1,1]]}},"URL":"https:\/\/doi.org\/10.1215\/00294527-2143952","relation":{},"ISSN":["0029-4527"],"issn-type":[{"value":"0029-4527","type":"print"}],"subject":[],"published":{"date-parts":[[2013,1,1]]}}}