On Sums of Generating Sets in ℤ2n

Let A and B be two affinely generating sets of ℤ2n. As usual, we denote their Minkowski sum by A+B. How small can A+B be, given the cardinalities of A and B? We give a tight answer to this question. Our bound is attained when both A and B are unions of cosets of a certain subgroup of ℤ2n. These cosets are arranged as Hamming balls, the smaller of which has radius 1. By similar methods, we re-prove the Freiman-Ruzsa theorem in ℤ2n, with an optimal upper bound. Denote by F(K) the maximal spanning constant |a€ˆ A a€‰|/|A| over all subsets A ⊆ ℤ2n with doubling constant |A+A|/|A| ≤ K. We explicitly calculate F(K), and in particular show that 4K/4K ≤ F(K)⋠(1+o(1)) ≤ 4K/2K. This improves the estimate F(K) = poly(K)4K, found recently by Green and Tao [17] and by Konyagin [23].

[1]  Albrecht Pfjster,et al.  Zur Darstellung von -1 Als Summe von Quadraten in Einem Körper , 1965 .

[2]  Michel Kervaire,et al.  Old and new formulas for the Hopf-Stiefel and related functions , 2005 .

[3]  Harold Davenport,et al.  On the Addition of Residue Classes , 1935 .

[4]  Eduard L. Stiefel,et al.  Über Richtungsfelder in den projektiven Räumen und einen Satz aus der reellen Algebra , 1940 .

[5]  I M R E Z. RUZSA.  Sum of sets in several dimensions , 1994, Comb..

[6]  Ben Green,et al.  Sets with Small Sumset and Rectification , 2004, math/0403338.

[7]  Sergei Konyagin.  On the Freiman theorem in finite fields , 2008 .

[8]  L. H. Harper.  Optimal numberings and isoperimetric problems on graphs , 1966 .

[9]  Gyula O. H. Katona,et al.  Intersection theorems for systems of finite sets , 1964 .

[10]  A. Vosper,et al.  The Critical Pairs of Subsets of a Group of Prime Order , 1956 .

[11]  Béla Bollobás,et al.  Sums in the grid , 1996, Discret. Math..

[12]  Daniel B. Shapiro,et al.  Compositions of Quadratic Forms , 2000 .

[13]  Alain Plagne,et al.  On Small Sumsets in (ℤ/2ℤ)n , 2004 .

[14]  Imre Z. Ruzsa,et al.  Generalized arithmetical progressions and sumsets , 1994 .

[15]  Gyula O. H. Katona,et al.  The Hamming-sphere has minimum boundary , 1975 .

[16]  Alain Plagne,et al.  Optimally small sumsets in finite abelian groups , 2003 .

[17]  Gilles Zémor.  An Extremal Problem Related to the Covering Radius of Binary Codes , 1991, Algebraic Coding.

[18]  Alain Plagne,et al.  On the subgroup generated by a small doubling binary set , 2003, Eur. J. Comb..

[19]  Vsevolod F. Lev.  Critical pairs in abelian groups and Kemperman's structure theorem , 2005 .

[20]  Gilles Zémor.  Subset sums in binary spaces , 1992, Eur. J. Comb..

[21]  Peter Frankl,et al.  A new short proof for the Kruskal-Katona theorem , 1984, Discret. Math..

[22]  Michel Kervaire,et al.  Minimal sumsets in infinite abelian groups , 2005 .

[23]  Ben Green,et al.  Freiman's Theorem in Finite Fields via Extremal Set Theory , 2009, Comb. Probab. Comput..

[24]  Vsevolod F. Lev,et al.  Generating binary spaces , 2003, J. Comb. Theory, Ser. A.

[25]  P. Erdös,et al.  Intersection Theorems for Systems of Sets , 1960 .

[26]  A. Cauchy.  Oeuvres complètes: Recherches sur les nombres , 2009 .

[27]  Michel Kervaire,et al.  Sumsets in Vector Spaces over Finite Fields , 1998 .

[28]  Peter Frankl.  A lower bound on the size of a complex generated by an antichain , 1989, Discret. Math..

[29]  Imre Z. Ruzsa,et al.  Arithmetical progressions and the number of sums , 1992 .

[30]  Sergey Yuzvinsky,et al.  Orthogonal pairings of Euclidean spaces. , 1981 .

[31]  G. Freiman.  Foundations of a Structural Theory of Set Addition , 2007 .

[32]  T. Sanders,et al.  A Note on Freĭman's Theorem in Vector Spaces , 2006, Combinatorics, Probability and Computing.

[33]  R. Schneider.  Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition , 1993 .

[34]  Hansheng Diao,et al.  Freiman–Ruzsa-type theory for small doubling constant , 2008, Mathematical Proceedings of the Cambridge Philosophical Society.

[35]  Emanuele Viola,et al.  Selected Results in Additive Combinatorics: An Exposition , 2007, Theory Comput..

[36]  Richard J. Gardner,et al.  A Brunn-Minkowski inequality for the integer lattice , 2001 .

[37]  H. Hopf,et al.  Ein toplogischer Beitrag zur reellen Algebra , 1940 .