Restricted sumset

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

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 field $\Bbb Z/p\Bbb Z$ we have the inequality

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

Erdős–Heilbronn conjecture

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 $\Bbb Z/p\Bbb Z$. This was first confirmed by J. A. Dias da Silva and Y. O. Hamidoune in 1994^[1] 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 $p(F)=\infty$ if F is of characteristic 0. Various extensions of this result were given by Noga Alon, M. B. Nathanson and I. Ruzsa in 1996,^[2] Q. H. Hou and Zhi-Wei Sun in 2002,^[3] and G. Karolyi in 2004.^[4]

Combinatorial Nullstellensatz

A powerful tool in the study of lower bounds for cardinalities of various restricted sumsets is the following fundamental principle: the combinatorial Nullstellensatz.^[5] 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,^[6] and developed by Alon, Nathanson and Ruzsa in 1995-1996,^[2] and reformulated by Alon in 1999.^[5]

References

1. ^ 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. 
2. ^ ^{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. MR1373563. http://www.math.tau.ac.il/~nogaa/PDFS/anrf3.pdf. 
3. ^ Hou, Qing-Hu; Sun, Zhi-Wei (2002). "Restricted sums in a field". Acta Arithmetica 102 (3): 239–249. doi:10.4064/aa102-3-3. MR1884717. 
4. ^ Károlyi, Gyula (2004). "The Erdős–Heilbronn problem in abelian groups". Israel Journal of Mathematics 139: 349–359. doi:10.1007/BF02787556. MR2041798. 
5. ^ ^{a b} Alon, Noga (1999). "Combinatorial Nullstellensatz". Combinatorics, Probability and Computing 8 (1–2): 7–29. doi:10.1017/S0963548398003411. MR1684621. http://www.math.tau.ac.il/~nogaa/PDFS/null2.pdf. 
6. ^ Alon, Noga; Tarsi, Michael (1989). "A nowhere-zero point in linear mappings". Combinatorica 9 (4): 393–395. doi:10.1007/BF02125351. MR1054015. 

External links

• Sun, Zhi-Wei (2006). "An additive theorem and restricted sumsets". Math. Res. Lett. , no. 15 (6): 1263–1276. arXiv:math.CO/0610981. 
• Zhi-Wei Sun: On some conjectures of Erdős-Heilbronn, Lev and Snevily (PDF), a survey talk.
• Weisstein, Eric W., "Erdos-Heilbronn Conjecture" from MathWorld.