Abstract
We obtain critical pair theorems for subsets S and T of an abelian group such that |S + T| ≤ |S| + |T|. We generalize some results of Chowla, Vosper, Kemperman and a more recent result due to Rødseth and one of the authors.
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N. Alon, M. B. Nathanson and I. Z. Ruzsa: The polynomial method and restricted sums of congruence classes, J. Number Theory 56 (1996), 404–417.
A. L. Cauchy: Recherches sur les nombres, J. Ecole Polytechnique 9 (1813), 99–116.
S. Chowla: A theorem on the addition of residue classes: applications to the number Γ(k) in Waring’s problem, Proc. Indian Acad. Sci. 2 (1935), 242–243.
H. Davenport: On the addition of residue classes, J. London Math. Soc. 10 (1935), 30–32.
J.-M. Deshouillers and G. A. Freiman: A step beyond Kneser’s Theorem for abelian finite groups, Proc. London Math. Soc. (3) 86(1) (2003), 1–28.
B. Green and I. Z. Ruzsa: Sets with small sumset and rectification, Bull. London Math. Soc. 38 (2006), 43–52.
Y. O. Hamidoune: On the connectivity of Cayley digraphs, Europ. J. Combinatorics 5 (1984), 309–312.
Y. O. Hamidoune: An isoperimetric method in Additive Theory, J. Algebra 179 (1996), 622–630.
Y. O. Hamidoune: Some results in additive number theory I: The critical pair theory, Acta Arithmetica 96 (2000), 97–119.
Y. O. Hamidoune and Ø. J. Røseth: An inverse theorem modulo p, Acta Arithmetica 92 (2000), 251–262.
Y. O. Hamidoune, A. S. Lladó and O. Serra: On subsets with a small product in torsion-free groups, Combinatorica 18(4) (1998), 529–540.
Y. O. Hamidoune: On small subset product in a group. Structure Theory of setaddition; Astérisque no. 258(xiv–xv) (1999), 281–308.
Y. O. Hamidoune and A. Plagne: A new critical pair theorem applied to sum-free sets in abelian groups, Commentarii Mathematici Helvetici 79(1) (2004), 183–207.
Y. O. Hamidoune, O. Serra and G. Zémor: On the critical pair theory in ℤ/pℤ, Acta Arithmetica 121 (2006), 99–115.
G. A. Freiman: Foundations of a structural theory of set addition, Transl. Math. Monographs 37, Amer. Math. Soc., Providence, RI, 1973.
Gy. Károlyi: An inverse theorem for the restricted set addition in abelian groups, J. Algebra 290 (2005), 557–593.
Gy. Károlyi: Cauchy-Davenport theorem in group extensions, Enseign. Math. (2) 51(3–4) (2005), 239–254.
J. H. B. Kemperman: On small sumsets in an abelian group, Acta Math. 103 (1960), 63–88.
M. Kneser: Summenmengen in lokalkompakten abelschen Gruppen, Math. Zeit. 66 (1956), 88–110.
H. B. Mann: An addition theorem of abelian groups for sets of elements, Proc. Amer. Math. Soc. 4 (1953), 423.
M. B. Nathanson: Additive Number Theory; Inverse problems and the geometry of sumsets, Grad. Texts in Math. 165, Springer, 1996.
O. Serra and G. Zémor: On a generalization of a theorem by Vosper, Integers Electr. J. Comb. Num. Th. 0(200), A10 (electronic). http://www.integersejcnt.org/vol0.html
T. Tao and V. H. Vu: Additive Combinatorics, Cambridge Studies in Advanced Mathematics 105 (2006), Cambridge University Press.
G. Vosper: The critical pairs of subsets of a group of prime order, J. London Math. Soc. 31 (1956), 200–205.
G. Zémor: A generalization to noncommutative groups of a theorem of Mann, Discrete Math. 126(1–3) (1994), 365–372.
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Supported by the Spanish Research Council under project MTM2005-08990-C02-01 and by the Catalan Research Council under project 2005SGR00256.
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Hamidoune, Y.O., Serra, O. & Zémor, G. On the critical pair theory in abelian groups: Beyond Chowla’s Theorem. Combinatorica 28, 441–467 (2008). https://doi.org/10.1007/s00493-008-2262-8
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DOI: https://doi.org/10.1007/s00493-008-2262-8