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

$P(x_{1},\ldots ,x_{n})=\prod _{{1\leq i<j\leq n}}(x_{j}-x_{i}),$

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 prime order cyclic group Z/pZ we have the inequality^[1][2]

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

We may use this to deduce the Erdős–Ginzburg–Ziv theorem: given any sequence of 2n−1 elements in Z/n, there are n elements 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 nonzero 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

The Erdős–Heilbronn conjecture posed by Paul Erdős and Hans Heilbronn in 1964 states that $|2^{\wedge }A|\geq \min\{p,2|A|-3\}$ 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

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

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

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

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. Alon, Noga; Nathanson, Melvyn B.; Ruzsa, Imre (1996). "The polynomial method and restricted sums of congruence classes" (PDF). 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. Alon, Noga (1999). "Combinatorial Nullstellensatz" (PDF). 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.

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. "Erdős-Heilbronn Conjecture". MathWorld.