Small doubling in $m$-Engel groups

Document Type : Proceedings of the conference "Engel conditions in groups" - Bath - UK - 2019


1 Department of Mathematics, University of Salerno

2 University of Salerno


We study some inverse problems of small doubling type in the class of $m$-Engel groups‎. ‎In particular we investigate the structure of a finite subset $S$ of a torsion-free $m$-Engel group if $|S^2| = 2|S|+b$‎, ‎where $0 \leq b \leq |S|-4$‎, ‎for some values of $b$‎.


Main Subjects

[1] A. Abdollahi and F. Jafari, Cardinality of product sets in torsion-free groups and applications in group algebras, J.
Algebra Appl., 19 (2020) pp. 24.
[2] Y. Bilu, Structure of sets with small sumset, Ast´erisque, 258 (1999) 77–108.
[3] V. V. Bludov, A. M. W. Glass, V. M. Kopitov and N. Ya Medvedev, Unsolved problems in ordered and orderable
[4] R. Botto Mura and A. Rhemtulla, Orderable groups, Lecture Notes in Pure and Applied Mathematics, Vol. 27,
Marcel Dekker, Inc., New York-Basel, 1977.
[5] L. V. Brailovsky and G. A. Freiman, On a product of finite subsets in a torsion-free group, J. Algebra, 130 (1990)
[6] E. Breuillard, B. Green and T. Tao, Small doubling in groups, Erd˝os centennial, 129–151, Bolyai Soc. Math. Stud.,
Vol. 25, Jnos Bolyai Math. Soc., Budapest, 2013.
[7] K. J. B¨or¨oczky, P. P. P´alfy and O. Serra, On the cardinality of sumsets in torsion-free groups, Bull. London Math.
Soc., 44 (2012) 1034–1041.
[8] W. Carter, Non-unique product on two generators, Master Thesis, Virginia Tech., 2003.
[9] J.-M. Deshouillers, F. Hennecart and A. Plagne, On small sumsets in (Z/2Z)n, Combinatorica, 24 (2004) 53–68.
[10] G. A. Freiman, On the addition of finite sets. I., Izv. Vyss. Ucebn. Zaved. Matematika, 6 (13) (1959) 202–213.
[11] G. A. Freiman, Inverse problems of additive number theory. IV: On the addition of finite sets. II, (Russian) Elabuz.
Gos. Ped. Inst. Ucen. Zap., 8 (1960) 72–116.
[12] G. A. Freiman, Foundations of a structural theory of set addition, Translated from the Russian. Translations of
Mathematical Monographs, Vol 37. American Mathematical Society, Providence, R. I., 1973.
[13] G. A. Freiman, Structure Theory of Set Addition, Ast´erisque, 258 (1999) 1–33.
[14] G. A. Freiman, M. Herzog, P. Longobardi and M. Maj, Small doubling in ordered groups, J. Austral. Math. Soc.,
96 (2014) 316–325.
[15] G. A. Freiman, M. Herzog, P. Longobardi, M. Maj and Y. V. Stanchescu, Direct and inverse problems in additive
number theory and in non-abelian group theory, European J. Combin., 40 (2014) 42–54.
[16] G. A. Freiman, M. Herzog, P. Longobardi, M. Maj and Y. V. Stanchescu, Some inverse problems in group theory,
Note Mat., 34 (2014) 89–104.
[17] G.A. Freiman, M. Herzog, P. Longobardi, M. Maj, Y.V. Stanchescu, A small doubling structure theorem in a
Baumslag-Solitar group, European J. Combin., 44 (2015) 106–124.
[18] G. A. Freiman, M. Herzog, P. Longobardi, M. Maj and Y. V. Stanchescu, Small doubling in nilpotent groups of
class 2, European J. Comb., 67 (2018) 87–95.
[19] G. A. Freiman, M. Herzog, P. Longobardi, M. Maj, A. Plagne, D. J. S. Robinson and Y. V. Stanchescu, On the
structure of subsets of an orderable group with some small doubling properties, J. Algebra, 445 (2016) 307–326.
[20] G. A. Freiman, M. Herzog, P. Longobardi, M. Maj, A. Plagne and Y. V. Stanchescu, Small doubling in ordered
groups: Generators and structure, Groups Geom. Dyn., 11 (2017) 585–612.
[21] G. A. Freiman and B. M. Schein, Group and semigroup theoretic considerations inspired by inverse problems of
the additive number theory, in Lecture Notes in Math., 1320, Springer-Verlag, Berlin-Heidelberg-New York, 1988
[22] A. Geroldinger and I. Z. Ruzsa, Combinatorial Number Theory and Additive Group Theory, Birkh¨auser, BaselBoston-Berlin, 2009.
[23] A. M. W. Glass, Partially ordered groups, Series in Algebra, vol. 7, World Scientific Publishing Co., Inc., River
Edge, NJ, 1999.
[24] B. Green, What is ... an approximate group?, Notices Amer. Math. Soc., 59 (2012) 655–656.
[25] B. Green and I. Z. Ruzsa, Freiman’s theorem in an arbitrary abelian group, J. London Math. Soc., 75 (2007)
[26] Y. O. Hamidoune, An application of connectivity theory in graphs to factorizations of elements in groups, European
J. Combin., 2 (1981) 349–355.
[27] Y. O. Hamidoune, An isoperimetric method in additive theory, J. Algebra, 179 (1996) 622–630.
[28] Y. O. Hamidoune, Some additive applications of the isoperimetric approach, Ann. Inst. Fourier (Grenoble), 58
(2008) 2007–2036.
[29] Y. O. Hamidoune, The isoperimetric method, Combinatorial number theory additive group theory, Adv. Courses
Math. CRM Barcelona, Birkh¨auser, Basel, 2009 241–252.
[30] Y. O. Hamidoune, A. S. Llad´o and O. Serra, On subsets with small product in torsion-free groups, Combinatorica,
18 (1998) 529–540.
[31] Y. O. Hamidoune and A. Plagne, A generalization of Freiman’s 3k − 3 Theorem, Acta Arith., 103 (2002) 147–156.
[32] Y. O. Hamidoune and A. Plagne, A multiple set version of the 3k − 3 theorem, Rev. Mat. Iberoam., 21 (2005)
[33] M. Herzog, P. Longobardi and M. Maj, Some results on products of finite subsets in groups, C. M. Campbell, M. R.
Quick, E. F. Robertson and C. M. Roney-Dougal, Groups St Andrews 2013, London Math. Soc. Lecture Note Ser.
422 (2015), Cambridge University Press, 286–305.
[34] J. H. B. Kemperman, On complexes in a semigroup, Indag. Math., 18 (1956) 247–254.
[35] Y. Kim and A. H. Rhemtulla, Orderable groups satisfying an Engel condition, Ordered Algebraic Structures: Proc.
of the 1991 Conrad Conference (ed. J. Martinez and W.Ch. Holland), Kluwer Academic Publ., Dordrecht, 1992
[36] M. Kneser, Absch¨atzung der asymptotischen Dichte von Summenmengen, Math. Z., 58 (1953) 459–484.
[37] L. Kurdachenko, P. Longobardi and M. Maj, Groups with numerical restrictions on minimal generating sets, to
appear in International Journal of Group Theory.
[38] V. F. Lev and P. Y. Smeliansky, On addition of two distinct sets of integers, Acta Arith., 70 (1995) 85–91.
[39] M. B. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Springer, 1996.
[40] S. D. Promislow, A simple example of a torsion-free, non unique product group, Bull. London Math. Soc., 20 (1988)
[41] E. Rips and Y. Segev, Torsion-free groups without unique product property, J. Algebra, 108 (1987) 116–126.
[42] D. J. S. Robinson, A course in the theory of groups, 2nd ed., Springer, 1996.
[43] I. Z. Ruzsa, An analog of Freiman’s theorem in groups, Ast´erisque, 258 (1999) 323–326.
[44] I. Z. Ruzsa, Sumsets and structure, Ast´erisque, Combinatorial number theory and additive group theory, Adv.
Courses Math. CRM Barcelona, Birkh¨auser Verlag, Basel, 2009 87–210.
[45] T. Sanders, The structure theory of set addition revisited, Bull. Amer. Math. Soc., 501 (2013) 93–127.
[46] Y. V. Stanchescu, On addition of two distinct sets of integers, Acta Arith., 75 (1996) 191–194.
[47] Y. V. Stanchescu, On the structure of sets with small doubling property on the plane. I, Acta Arith., 83 (1998)
[48] Y. V. Stanchescu, The structure of d-dimensional sets with small sumset, J. Number Theory, 30 (2010) 289–303.
[49] A. Strojnowski, A note on u.p. groups, Comm. Algebra, 8 (1980) 231–234.
[50] T. Tao and Van H. Vu, Additive combinatorics, Cambridge Stud. Adv. Math., 105, Cambridge University Press,
[51] T. Tao, Product set estimates for non-commutative groups, Combinatorica, 28 (2008) 547–594.
[52] G. Traustason Engel groups, Groups St Andrews 2009 in Bath, Vol. II, London Math. Soc. Lecture Note Ser., 388
(2011), Cambridge University Press, 520–550.