Restricted sumset

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In additive number theory and combinatorics, a restricted sumset has the form

S=\{a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n \ \mathrm{and}\ P(a_1,\ldots,a_n)\not=0\},

where  A_1,\ldots,A_n are finite nonempty subsets of a field F and P(x_1,\ldots,x_n) is a polynomial over F.

When P(x_1,\ldots,x_n)=1, S is the usual sumset A_1+\cdots+A_n which is denoted by nA if A_1=\cdots=A_n=A; when

P(x_1,\ldots,x_n)=\prod_{1\le i<j\le n}(x_j-x_i),

S is written as A_1\dotplus\cdots\dotplus A_n which is denoted by n^{\wedge} A if A_1=\cdots=A_n=A. Note that |S| > 0 if and only if there exist a_1\in A_1,\ldots,a_n\in A_n with P(a_1,\ldots,a_n)\not=0.

Cauchy–Davenport theorem[edit]

The Cauchy–Davenport theorem named after Augustin Louis Cauchy and Harold Davenport asserts that for any prime p and nonempty subsets A and B of the prime order cyclic group Z/pZ we have the inequality[1][2]

|A+B|\ge\min\{p,\ |A|+|B|-1\}.\,

We may use this to deduce the Erdős-Ginzburg-Ziv theorem: given any 2n−1 elements of Z/n, there is a non-trivial subset that sums to zero modulo n. (Here n does not need to be prime.)[3][4]

A direct consequence of the Cauchy-Davenport theorem is: Given any set S of p−1 or more elements, not necessarily distinct, of Z/pZ, every element of Z/pZ can be written as the sum of the elements of some subset (possibly empty) of S.[5]

Kneser's theorem generalises this to finite abelian groups.[6]

Erdős–Heilbronn conjecture[edit]

The Erdős–Heilbronn conjecture posed by Paul Erdős and Hans Heilbronn in 1964 states that |2^\wedge A|\ge\min\{p,2|A|-3\} if p is a prime and A is a nonempty subset of the field Z/pZ.[7] This was first confirmed by J. A. Dias da Silva and Y. O. Hamidoune in 1994[8] who showed that

|n^\wedge A|\ge\min\{p(F),\ n|A|-n^2+1\},

where A is a finite nonempty subset of a field F, and p(F) is a prime p if F is of characteristic p, and p(F) = ∞ if F is of characteristic 0. Various extensions of this result were given by Noga Alon, M. B. Nathanson and I. Ruzsa in 1996,[9] Q. H. Hou and Zhi-Wei Sun in 2002,[10] and G. Karolyi in 2004.[11]

Combinatorial Nullstellensatz[edit]

A powerful tool in the study of lower bounds for cardinalities of various restricted sumsets is the following fundamental principle: the combinatorial Nullstellensatz.[12] Let f(x_1,\ldots,x_n) be a polynomial over a field F. Suppose that the coefficient of the monomial x_1^{k_1}\cdots x_n^{k_n} in f(x_1,\ldots,x_n) is nonzero and k_1+\cdots+k_n is the total degree of f(x_1,\ldots,x_n). If A_1,\ldots,A_n are finite subsets of F with |A_i|>k_i for i=1,\ldots,n, then there are a_1\in A_1,\ldots,a_n\in A_n such that f(a_1,\ldots,a_n)\not = 0 .

The method using the combinatorial Nullstellensatz is also called the polynomial method. This tool was rooted in a paper of N. Alon and M. Tarsi in 1989,[13] and developed by Alon, Nathanson and Ruzsa in 1995-1996,[9] and reformulated by Alon in 1999.[12]

References[edit]

  1. ^ Nathanson (1996) p.44
  2. ^ Geroldinger & Ruzsa (2009) pp.141–142
  3. ^ Nathanson (1996) p.48
  4. ^ Geroldinger & Ruzsa (2009) p.53
  5. ^ Wolfram's MathWorld, Cauchy-Davenport Theorem, http://mathworld.wolfram.com/Cauchy-DavenportTheorem.html, accessed 20 June 2012.
  6. ^ Geroldinger & Ruzsa (2009) p.143
  7. ^ Nathanson (1996) p.77
  8. ^ Dias da Silva, J. A.; Hamidoune, Y. O. (1994). "Cyclic spaces for Grassman derivatives and additive theory". Bulletin of the London Mathematical Society 26 (2): 140–146. doi:10.1112/blms/26.2.140. 
  9. ^ a b Alon, Noga; Nathanson, Melvyn B.; Ruzsa, Imre (1996). "The polynomial method and restricted sums of congruence classes". Journal of Number Theory 56 (2): 404–417. doi:10.1006/jnth.1996.0029. MR 1373563. 
  10. ^ Hou, Qing-Hu; Sun, Zhi-Wei (2002). "Restricted sums in a field". Acta Arithmetica 102 (3): 239–249. doi:10.4064/aa102-3-3. MR 1884717. 
  11. ^ Károlyi, Gyula (2004). "The Erdős–Heilbronn problem in abelian groups". Israel Journal of Mathematics 139: 349–359. doi:10.1007/BF02787556. MR 2041798. 
  12. ^ a b Alon, Noga (1999). "Combinatorial Nullstellensatz". Combinatorics, Probability and Computing 8 (1–2): 7–29. doi:10.1017/S0963548398003411. MR 1684621. 
  13. ^ Alon, Noga; Tarsi, Michael (1989). "A nowhere-zero point in linear mappings". Combinatorica 9 (4): 393–395. doi:10.1007/BF02125351. MR 1054015. 
  • Geroldinger, Alfred; Ruzsa, Imre Z., eds. (2009). Combinatorial number theory and additive group theory. Advanced Courses in Mathematics CRM Barcelona. Elsholtz, C.; Freiman, G.; Hamidoune, Y. O.; Hegyvári, N.; Károlyi, G.; Nathanson, M.; Solymosi, J.; Stanchescu, Y. With a foreword by Javier Cilleruelo, Marc Noy and Oriol Serra (Coordinators of the DocCourse). Basel: Birkhäuser. ISBN 978-3-7643-8961-1. Zbl 1177.11005. 
  • Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics 165. Springer-Verlag. ISBN 0-387-94655-1. Zbl 0859.11003. 

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