# Restricted sumset

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

where are finite nonempty subsets of a field *F* and is a polynomial over *F*.

When , *S* is the usual sumset which is denoted by *nA* if ; when

*S* is written as which is denoted by if . Note that |*S*| > 0 if and only if there exist with .

## 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**/*p***Z** we have the inequality[1][2]

We may use this to deduce the Erdős–Ginzburg–Ziv theorem: given any sequence of 2*n*−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**/*p***Z**, every element of **Z**/*p***Z** can be written as the sum of the elements of some subset (possibly empty) of *S*.[5]

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

## Erdős–Heilbronn conjecture

The **Erdős–Heilbronn conjecture** posed by Paul Erdős and Hans Heilbronn in 1964 states that if *p* is a prime and *A* is a nonempty subset of the field **Z**/*p***Z**.[7] This was first confirmed by J. A. Dias da Silva and Y. O. Hamidoune in 1994[8]
who showed that

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 be a polynomial over a field *F*. Suppose that the coefficient of the monomial in is nonzero and is the total degree of . If are finite subsets of *F* with for , then there are such that .

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]

## See also

## References

- Nathanson (1996) p.44
- Geroldinger & Ruzsa (2009) pp.141–142
- Nathanson (1996) p.48
- Geroldinger & Ruzsa (2009) p.53
- Wolfram's MathWorld, Cauchy-Davenport Theorem, http://mathworld.wolfram.com/Cauchy-DavenportTheorem.html, accessed 20 June 2012.
- Geroldinger & Ruzsa (2009) p.143
- Nathanson (1996) p.77
- Dias da Silva, J. A.; Hamidoune, Y. O. (1994). "Cyclic spaces for Grassmann derivatives and additive theory".
*Bulletin of the London Mathematical Society*.**26**(2): 140–146. doi:10.1112/blms/26.2.140. - 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. - Hou, Qing-Hu; Sun, Zhi-Wei (2002). "Restricted sums in a field".
*Acta Arithmetica*.**102**(3): 239–249. Bibcode:2002AcAri.102..239H. doi:10.4064/aa102-3-3. MR 1884717. - Károlyi, Gyula (2004). "The Erdős–Heilbronn problem in abelian groups".
*Israel Journal of Mathematics*.**139**: 349–359. doi:10.1007/BF02787556. MR 2041798. - Alon, Noga (1999). "Combinatorial Nullstellensatz" (PDF).
*Combinatorics, Probability and Computing*.**8**(1–2): 7–29. doi:10.1017/S0963548398003411. MR 1684621. - 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*.**15**(6): 1263–1276. arXiv:math.CO/0610981. Bibcode:2006math.....10981S. doi:10.4310/MRL.2008.v15.n6.a15. - Zhi-Wei Sun: On some conjectures of Erdős-Heilbronn, Lev and Snevily (PDF), a survey talk.
- Weisstein, Eric W. "Erdős-Heilbronn Conjecture".
*MathWorld*.