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References

Published online by Cambridge University Press:  31 October 2019

Matthew C. H. Tointon
Matthew C. H. Tointon
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University of Cambridge
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Introduction to Approximate Groups , pp. 198 - 201
Publisher: Cambridge University Press
Print publication year: 2019

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References

[1] Babai, L. and Seress, A., On the diameter of Cayley graphs of the symmetric group. J. Combin. Theory Ser. A, 49 (1988), 175–179.Google Scholar
[2] Balog, A. and Szemerédi, E., A statistical theorem of set addition. Combinatorica, 14(3) (1994), 263–268.Google Scholar
[3] Bass, H., The degree of polynomial growth of finitely generated nilpotent groups. Proc. Lond. Math. Soc. Ser. 3, 25(4) (1972), 603–614.CrossRefGoogle Scholar
[4] Benjamini, I. and Kozma, G., A resistance bound via an isoperimetric inequality. Combinatorica, 25(6) (2005), 645–650.CrossRefGoogle Scholar
[5] Billingsley, P., Probability and Measure, 3rd edn (New York: John Wiley & Sons, 1995).Google Scholar
[6]M. Björklund and T. Hartnick, Approximate lattices. Preprint, arXiv:1612.09246.Google Scholar
[7] Blichfeldt, H. F., A new principle in the geometry of numbers with some applications. Trans. Amer. Math. Soc., 15(3) (1914), 227–235.Google Scholar
[8] Bogolyubov, N. N., Sur quelques propriétés arithmétiques des presque-périodes. Ann. Chaire Math. Phys. Kiev, 4 (1939), 185–194.Google Scholar
[9] Bourgain, J. and Gamburd, A., Uniform expansion bounds for Cayley graphs of SL2(Fp). Ann. Math., 167(2) (2008), 625–642.Google Scholar
[10] Bourgain, J., Katz, N. and Tao, T., A sum-product estimate in finite fields, and applications. Geom. Funct. Anal., 14(1) (2004), 27–57.CrossRefGoogle Scholar
[11]E. Breuillard, A strong Tits alternative. Preprint, arXiv:0804.1395.Google Scholar
[12] Breuillard, E. and Green, B. J., Approximate groups I: The torsion-free nilpotent case. J. Inst. Math. Jussieu, 10(1) (2011), 37–57.CrossRefGoogle Scholar
[13] Breuillard, E. and Green, B. J., Approximate groups II: The solvable linear case. Q. J. Math., 62(3) (2011), 513–521.Google Scholar
[14] Breuillard, E. and Green, B. J., Approximate groups III: The unitary case. Turkish J. Math., 36(2) (2012), 199–215.Google Scholar
[15] Breuillard, E., Green, B. J. and Tao, T., Linear approximate groups (research announcement). Electron. Res. Announc. Math. Sci., 17 (2010), 57–67.Google Scholar
[16] Breuillard, E., Green, B. J. and Tao, T., Approximate subgroups of linear groups. Geom. Funct. Anal., 21(4) (2011), 774–819.Google Scholar
[17]E. Breuillard, B. J. Green and T. Tao, A note on approximate subgroups of GLn(C) and uniformly nonamenable groups. Preprint, arXiv:1101.2552.Google Scholar
[18] Breuillard, E., Green, B. J. and Tao, T., The structure of approximate groups. Publ. Math. IHES, 116(1) (2012), 115–221.Google Scholar
[19] Breuillard, E., Green, B. J. and Tao, T., A nilpotent Freiman dimension lemma. Eur. J. Combin., 34(8) (2013), 1287–1292.Google Scholar
[20] Breuillard, E. and Tointon, M. C. H., Nilprogressions and groups with moderate growth. Adv. Math., 289 (2016), 1008–1055.Google Scholar
[21] Cassels, J. W. S., An Introduction to the Geometry of Numbers, corrected reprint of the 1971 edition, Classics in Mathematics (Berlin: Springer-Verlag, 1997).Google Scholar
[22] Chang, M. C., A polynomial bound in Freiman’s theorem. Duke Math. J., 113(3) (2002), 399–419.Google Scholar
[23] Croot, E. and Sisask, O., A probabilistic technique for finding almost-periods of convolutions. Geom. Funct. Anal., 20(6) (2010), 1367–1396.Google Scholar
[24] Cwalina, K. and Schoen, T., A linear bound on the dimension in Green– Ruzsa’s theorem. J. Number Theory, 133 (2013), 1262–1269.CrossRefGoogle Scholar
[25] de la Harpe, P., Topics in Geometric Group Theory, Chicago Lectures in Mathematics (Chicago: University of Chicago Press, 2000).Google Scholar
[26] Freiman, G. A., Foundations of a Structural Theory of Set Addition, Translations of Mathematical Monographs, vol. 37 (Providence, RI: Amer. Math. Soc., 1973). Translated from the 1966 Russian version published by Kazan Gos. Ped. Inst.Google Scholar
[27] Freiman, G., Number-theoretic studies in the Markov spectrum and in the structural theory of set addition. In Groups and the Inverse Problems of Additive Number Theory (in Russian) (Moscow: Kalinin. Gos. Univ., 1973).Google Scholar
[28] Gill, N. and Helfgott, H. A., Growth in solvable subgroups of GLr(Z/pZ). Math. Ann., 360(1) (2014), 157–208.Google Scholar
[29] Gleason, A. M., Compact subgroups. Proc. Natl. Acad. Sci. USA, 37(9) (1951), 622–623.Google Scholar
[30] Gowers, W. T., A new proof of Szemerédi’s theorem for arithmetic progressions of length four. Geom. Funct. Anal., 8(3) (1998), 529–551.Google Scholar
[31] Gowers, W. T., A new proof of Szemerédi’s theorem. Geom. Funct. Anal., 11(3) (2001), 465–588.Google Scholar
[32]W. T. Gowers, A new way of proving sumset estimates. Available at https://gowers.wordpress.com/2011/02/10/a-new-way-of-proving-sumset-estimates/.Google Scholar
[33]B. J. Green, Approximate groups and their applications: Work of Bourgain, Gamburd, Helfgott and Sarnak, arXiv:0911.3354.Google Scholar
[34] Green, B. J. and Ruzsa, I. Z., Sets with small sumset and rectification. Bull. Lond. Math. Soc., 38(1) (2006), 43–52.Google Scholar
[35] Green, B. J. and Ruzsa, I. Z., Freiman’s theorem in an arbitrary abelian group. J. Lond. Math. Soc., 75(1) (2007), 163–175.Google Scholar
[36] Gromov, M., Groups of polynomial growth and expanding maps. Publ. Math. IHES, 53 (1981), 53–73.Google Scholar
[37] Guralnick, R. M., On a result of Schur. J. Algebra, 59 (1979), 302–310.Google Scholar
[38] Hall, M., The Theory of Groups (Providence, RI: Amer. Math. Soc./Chelsea, 1999).Google Scholar
[39] Helfgott, H. A., Growth and generation in SL2(Z/pZ). Ann. Math., 167(2) (2008), 601–623.Google Scholar
[40] Helfgott, H. A., Growth in SL3(Z/pZ). J. Eur. Math. Soc., 13(3) (2011), 761–851.Google Scholar
[41] Hoory, S., Linial, N. and Wigderson, A., Expander graphs and their applications. Bull. Amer. Math. Soc., 43 (2006), 439–561.Google Scholar
[42] Hrushovski, E., Stable group theory and approximate subgroups. J. Amer. Math. Soc., 25(1) (2012), 189–243.Google Scholar
[43] Lovett, S. and Regev, O.. A counterexample to a strong variant of the polynomial Freiman-Ruzsa conjecture, Discrete Anal. (2017), article no. 8.Google Scholar
[44] Mal’cev, A. I., On certain classes of infinite soluble groups. Mat. Sb., 28 (1951), 567–588 (in Russian). Amer. Math. Soc. Transl. Ser. 2, 45 (1956), 1–21.Google Scholar
[45] Petridis, G., New proofs of Plünnecke-type estimates for product sets in groups. Combinatorica, 32(6) (2012), 721–733.Google Scholar
[46] Petridis, G., Upper bounds on the cardinality of higher sumsets. Acta Arith., 158(4) (2013), 299–319.Google Scholar
[47] Plünnecke, H., Eine zahlentheoretische Anwendung der Graphentheorie. J. Reine Angew. Math., 243 (1970), 171–183.Google Scholar
[48]L. Pyber and E. Szabó, Growth in finite simple groups of Lie type. Research announcement, arXiv:1001.4556.Google Scholar
[49] Pyber, L. and Szabó, E., Growth in finite simple groups of Lie type. J. Amer. Math. Soc., 29(1) (2016), 95–146.Google Scholar
[50] Razborov, A. A., A product theorem in free groups. Ann. Math., 179(2) (2014), 405–429.Google Scholar
[51] Robinson, D. J. S., A Course in the Theory of Groups, 2nd edn, Graduate Texts in Mathematics (New York: Springer-Verlag, 1995).Google Scholar
[52] Ruzsa, I. Z., On the cardinality of A + A and A − A. In Combinatorics, Coll. Math. Soc. J. Bolyai, vol. 18, (Budapest: North-Holland / Bolyai Társulat, Budapest, 1978), 933–938.Google Scholar
[53] Ruzsa, I. Z., An application of graph theory to additive number theory. Scientia, Ser. A, 3 (1989), 97–109.Google Scholar
[54] Ruzsa, I. Z., Addendum to: An application of graph theory to additive number theory. Scientia, Ser. A, 4 (1990/91), 93–94.Google Scholar
[55] Ruzsa, I. Z., Generalized arithmetical progressions and sumsets. Acta Math. Hungar., 65(4) (1994), 379–388.Google Scholar
[56] Ruzsa, I. Z., An analog of Freiman’s theorem in groups. Astérisque, 258 (1999), 323–326.Google Scholar
[57] Safin, S. R., Powers of subsets of free groups. Mat. Sb., 202(11) (2011), 97–102. English translation in Sb. Math., 202(11–12) (2011), 1661–1666.Google Scholar
[58] Sanders, T., On a non-abelian Balog–Szemerédi-type lemma. J. Aust. Math. Soc., 89(1) (2010), 127–132.CrossRefGoogle Scholar
[59] Sanders, T., On the Bogolyubov–Ruzsa lemma. Anal. PDE, 5(3) (2012), 627–655.Google Scholar
[60] Solymosi, J., On sum-sets and product-sets of complex numbers. J. Théor. Nombres Bordeaux, 17 (2005), 921–924.CrossRefGoogle Scholar
[61] Szemerédi, E., On sets of integers containing no k elements in arithmetic progression. Acta Arith., 27 (1975), 199–245.Google Scholar
[62] Tao, T., Product set estimates for non-commutative groups. Combinatorica, 28(5) (2008), 547–594.Google Scholar
[63] Tao, T., The sum-product phenomenon in arbitrary rings. Contrib. Discrete Math., 4(2) (2009), 59–82.Google Scholar
[64]T. Tao, An elementary non-commutative Freiman theorem. Available at https://terrytao.wordpress.com/2009/11/10/an-elementary-non-commutative-freiman-theorem/Google Scholar
[65] Tao, T., Freiman’s theorem for solvable groups. Contrib. Discrete Math., 5(2) (2010), 137–184.Google Scholar
[66] Tao, T., Expansion in Finite Simple Groups of Lie Type, Graduate Studies in Mathematics, vol. 164 (Providence, RI: Amer. Math. Soc., 2015).Google Scholar
[67] Tao, T., Inverse theorems for sets and measures of polynomial growth. Q. J. Math., 68(1) (2017), 13–57.Google Scholar
[68] Tao, T. and Vu, V. H., Additive Combinatorics, Cambridge Studies in Advanced Mathematics, vol. 105 (Cambridge: Cambridge University Press, 2006).Google Scholar
[69] Tessera, R. and Tointon, M. C. H., Properness of nilprogressions and the persistence of polynomial growth of given degree. Discrete Anal. (2018), article no. 17.Google Scholar
[70]R. Tessera and M. C. H. Tointon, Sharp relations between volume growth, isoperimetry and resistance in vertex-transitive graphs. In preparation.Google Scholar
[71] Tits, J., Free subgroups in linear groups. J. Algebra, 20 (1972), 250–270.Google Scholar
[72] Tointon, M. C. H., Freiman’s theorem in an arbitrary nilpotent group. Proc. Lond. Math. Soc. Ser. 3, 109 (2014), 318–352.Google Scholar
[73] Tointon, M. C. H., Approximate subgroups of residually nilpotent groups. Math. Ann., 374 (2019), 499–515.Google Scholar
[74]M. C. H. Tointon, Polylogarithmic bounds in the nilpotent Freiman theorem. Preprint, arXiv:1812.06735. To appear in Math. Proc. Camb. Phil. Soc.Google Scholar
[75] Wehrfritz, B. A. F., Infinite Linear Groups (Berlin: Springer-Verlag, 1973).Google Scholar

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