Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T08:08:47.288Z Has data issue: false hasContentIssue false
  • English
  • Français

On finite sets of small tripling or small alternation in arbitrary groups

Part of: Sequences and sets Model theory Representation theory of groups

Published online by Cambridge University Press:  30 June 2020

Gabriel Conant
Gabriel Conant*
Affiliation:
Department of Pure Mathematics & Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, UK
*
Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove Bogolyubov–Ruzsa-type results for finite subsets of groups with small tripling, |A3| ≤ O(|A|), or small alternation, |AA−1A| ≤ O(|A|). As applications, we obtain a qualitative analogue of Bogolyubov’s lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularity lemma for sets of bounded VC-dimension in finite groups of bounded exponent. The latter result generalizes the abelian case, due to Alon, Fox and Zhao, and gives a quantitative version of previous work of the author, Pillay and Terry.

MSC classification

Primary: 20D60: Arithmetic and combinatorial problems
Secondary: 11B30: Arithmetic combinatorics; higher degree uniformity 03C20: Ultraproducts and related constructions
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 6 , November 2020 , pp. 807 - 829
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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

Alekseev, M. A., Glebskiĭ, L. Y. and Gordon, E. I. (1999) On approximations of groups, group actions and Hopf algebras. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 256 224–262.Google Scholar
Alon, N., Fischer, E. and Newman, I. (2007) Efficient testing of bipartite graphs for forbidden induced subgraphs. SIAM J. Comput. 37 959–976.CrossRefGoogle Scholar
Alon, N., Fox, J. and Zhao, Y. (2019) Efficient arithmetic regularity and removal lemmas for induced bipartite patterns. Discrete Anal., Paper no. 3, 14 MR3943117.Google Scholar
Babai, L., Goodman, A. J. and Pyber, L. (1997) Groups without faithful transitive permutation representations of small degree. J. Algebra 195 1–29.CrossRefGoogle Scholar
Bogolioùboff, N. (1939) Sur quelques propriétés arithmétiques des presque-périodes. Ann. Chaire Phys. Math. Kiev 4 185–205.Google Scholar
Bourgain, J. (1999) On triples in arithmetic progression. Geom. Funct. Anal. 9 968–984.CrossRefGoogle Scholar
Breuillard, E., Green, B. and Tao, T. (2012) The structure of approximate groups. Publ. Math. Inst. Hautes Études Sci. 116 115–221.CrossRefGoogle Scholar
Collins, M. J. (2007) On Jordan’s theorem for complex linear groups. J. Group Theory 10 411–423.CrossRefGoogle Scholar
Conant, G., Pillay, A. and Terry, C. (2020) A group version of stable regularity. Math. Proc. Cambridge Philos. Soc., (2) 168 405–413. MR4064112.CrossRefGoogle Scholar
Conant, G., Pillay, A. and Terry, C. (2018) Structure and regularity for subsets of groups with finite VC-dimension.arXiv:1802.04246Google Scholar
Croot, E. and Sisask, O. (2010) A probabilistic technique for finding almost-periods of convolutions. Geom. Funct. Anal. 20 1367–1396.CrossRefGoogle Scholar
Freman, G. A. (1973) Foundations of a Structural Theory of Set Addition, Vol. 37 of Translations of Mathematical Monographs, AMS.Google Scholar
Freiman, G. A. (1987) What is the structure of K if K+K is small? In Number Theory (New York, 1984–1985), Vol. 1240 of Lecture Notes in Mathematics, pp. 109–134, Springer.CrossRefGoogle Scholar
Gowers, W. T. (2008) Quasirandom groups. Combin. Probab. Comput. 17 363–387.CrossRefGoogle Scholar
Green, B. (2005) A Szemerédi-type regularity lemma in abelian groups, with applications. Geom. Funct. Anal. 15 340–376.CrossRefGoogle Scholar
Green, B. and Ruzsa, I. Z. (2007) Freiman’s theorem in an arbitrary abelian group. J. London Math. Soc. (2) 75 163–175.CrossRefGoogle Scholar
Haussler, D. (1995) Sphere packing numbers for subsets of the Boolean n-cube with bounded Vapnik–Chervonenkis dimension. J. Combin. Theory Ser. A 69 217–232.CrossRefGoogle Scholar
Helfgott, H. A. (2008) Growth and generation in SL 2(Z/pZ). Ann. of Math. (2) 167 601–623.CrossRefGoogle Scholar
Hrushovski, E. (2012) Stable group theory and approximate subgroups. J. Amer. Math. Soc. 25 189–243.CrossRefGoogle Scholar
Hrushovski, E., Peterzil, Y. and Pillay, A. (2008) Groups, measures, and the NIP. J. Amer. Math. Soc. 21 563–596.CrossRefGoogle Scholar
Keisler, H. J. (1964) Ultraproducts and saturated models. Nederl. Akad. Wetensch. Proc. Ser. A 67 = Indag. Math. 26 178–186.CrossRefGoogle Scholar
Krupiński, K. and Pillay, A. (2016) Amenability, definable groups, and automorphism groups. Adv. Math. 345 1253–1299. MR3904280.CrossRefGoogle Scholar
Malliaris, M. and Shelah, S. (2014) Regularity lemmas for stable graphs. Trans. Amer. Math. Soc. 366 1551–1585.CrossRefGoogle Scholar
Massicot, J.-C. and Wagner, F. O. (2015) Approximate subgroups. J. École Polytech. Math. 2 55–64.CrossRefGoogle Scholar
Nikolov, N. and Pyber, L. (2011) Product decompositions of quasirandom groups and a Jordan type theorem. J. Eur. Math. Soc. 13 1063–1077.CrossRefGoogle Scholar
Nikolov, N., Schneider, J. and Thom, A. (2018) Some remarks on finitarily approximable groups. J. École Polytech. Math. 5 239–258.CrossRefGoogle Scholar
Pillay, A. (2004) Type-definability, compact Lie groups, and o-minimality. J. Math. Log. 4 147–162.CrossRefGoogle Scholar
Pillay, A. (2017) Remarks on compactifications of pseudofinite groups. Fund. Math. 236 193–200.CrossRefGoogle Scholar
Plünnecke, H. (1969) Eigenschaften und Abschätzungen von Wirkungsfunktionen, BMwF-GMD-22, Gesellschaft für Mathematik und Datenverarbeitung.Google Scholar
Ruzsa, I. Z. (1994) Generalized arithmetical progressions and sumsets. Acta Math. Hungar. 65 379–388.CrossRefGoogle Scholar
Ruzsa, I. Z. (1996) Sums of finite sets. In Number Theory (New York, 1991–1995), pp. 281–293, Springer.CrossRefGoogle Scholar
Sanders, T. (2010) On a nonabelian Balog–Szemerédi-type lemma. J. Aust. Math. Soc. 89 127–132.CrossRefGoogle Scholar
Sanders, T. (2012) On the Bogolyubov–Ruzsa lemma. Anal. PDE 5 627–655.CrossRefGoogle Scholar
Sisask, O. (2018) Convolutions of sets with bounded VC-dimension are uniformly continuous.arXiv:1802.02836Google Scholar
Tao, T. (2008) Product set estimates for non-commutative groups. Combinatorica 28 547–594.CrossRefGoogle Scholar
Tao, T. (2014) Hilbert’s Fifth Problem and Related Topics, Vol. 153 of Graduate Studies in Mathematics, AMS.Google Scholar
Tao, T. and Vu, V. (2006) Additive Combinatorics, Vol. 105 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.Google Scholar
Terry, C. and Wolf, J. (2019) Stable arithmetic regularity in the finite field model. Bull. Lond. Math. Soc. 51 70–88.CrossRefGoogle Scholar
Terry, C. and Wolf, J. (2018) Quantitative structure of stable sets in finite abelian groups. Trans. Amer. Math. Soc. accepted.Google Scholar