Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-22T21:03:35.247Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  31 October 2019

Matthew C. H. Tointon
Matthew C. H. Tointon
Affiliation:
University of Cambridge
Get access

Summary

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Introduction to Approximate Groups , pp. 198 - 201
Publisher: Cambridge University Press
Print publication year: 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×