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.

If $P$ is a constant non-zero function, for example $P(x_{1},\ldots ,x_{n})=1$ for any $x_{1},\ldots ,x_{n}$ , then $S$ 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 $\mathbb {Z} /p\mathbb {Z}$ we have the inequality[1][2][3]

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

where $A+B:=\{a+b{\pmod {p}}\mid a\in A,b\in B\}$ , i.e. we're using modular arithmetic. It can be generalised to arbitrary (not necessarily abelian) groups using a Dyson transform. If $A,B$ are subsets of a group $G$ , then[4]

|A+B|\geq \min\{p(G),\,|A|+|B|-1\}

where $p(G)$ is the size of the smallest nontrivial subgroup of $G$ (we set it to $1$ if there is no such subgroup).

We may use this to deduce the Erdős–Ginzburg–Ziv theorem: given any sequence of 2n−1 elements in the cyclic group $\mathbb {Z} /n\mathbb {Z}$ , there are n elements that sum to zero modulo n. (Here n does not need to be prime.)[5][6]

A direct consequence of the Cauchy-Davenport theorem is: Given any sequence S of p−1 or more nonzero elements, not necessarily distinct, of $\mathbb {Z} /p\mathbb {Z}$ , every element of $\mathbb {Z} /p\mathbb {Z}$ can be written as the sum of the elements of some subsequence (possibly empty) of S.[7]

Kneser's theorem generalises this to general abelian groups.[8]

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.[9] This was first confirmed by J. A. Dias da Silva and Y. O. Hamidoune in 1994[10] 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,[11] Q. H. Hou and Zhi-Wei Sun in 2002,[12] and G. Karolyi in 2004.[13]

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.[14] 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 $f(a_{1},\ldots ,a_{n})\not =0$ .

This tool was rooted in a paper of N. Alon and M. Tarsi in 1989,[15] and developed by Alon, Nathanson and Ruzsa in 1995–1996,[11] and reformulated by Alon in 1999.[14]

References

Nathanson (1996) p.44
Geroldinger & Ruzsa (2009) pp.141–142
Jeffrey Paul Wheeler (2012). "The Cauchy-Davenport Theorem for Finite Groups". arXiv:1202.1816 [math.CO].
DeVos, Matt (2016). "On a Generalization of the Cauchy-Davenport Theorem". Integers. 16.
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. S2CID 33387005.
Alon, Noga (1999). "Combinatorial Nullstellensatz" (PDF). Combinatorics, Probability and Computing. 8 (1–2): 7–29. doi:10.1017/S0963548398003411. MR 1684621. S2CID 209877602.
Alon, Noga; Tarsi, Michael (1989). "A nowhere-zero point in linear mappings". Combinatorica. 9 (4): 393–395. doi:10.1007/BF02125351. MR 1054015. S2CID 8208350.

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. Vol. 165. Springer-Verlag. ISBN 0-387-94655-1. Zbl 0859.11003.

External links

Weisstein, Eric W. "Erdős-Heilbronn Conjecture". MathWorld.

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[1] Nathanson (1996) p.44

[GR1412-2] Geroldinger & Ruzsa (2009) pp.141–142

[3] Jeffrey Paul Wheeler (2012). "The Cauchy-Davenport Theorem for Finite Groups". arXiv:1202.1816 [math.CO].

[4] DeVos, Matt (2016). "On a Generalization of the Cauchy-Davenport Theorem". Integers. 16.

[5] Nathanson (1996) p.48

[GR53-6] Geroldinger & Ruzsa (2009) p.53

[7] Wolfram's MathWorld, Cauchy-Davenport Theorem, http://mathworld.wolfram.com/Cauchy-DavenportTheorem.html, accessed 20 June 2012.

[GR143-8] Geroldinger & Ruzsa (2009) p.143

[9] Nathanson (1996) p.77

[10] 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.

[Alon1996-11] 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.

[12] 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.

[13] Károlyi, Gyula (2004). "The Erdős–Heilbronn problem in abelian groups". Israel Journal of Mathematics. 139: 349–359. doi:10.1007/BF02787556. MR 2041798. S2CID 33387005.

[Alon1999-14] Alon, Noga (1999). "Combinatorial Nullstellensatz" (PDF). Combinatorics, Probability and Computing. 8 (1–2): 7–29. doi:10.1017/S0963548398003411. MR 1684621. S2CID 209877602.

[15] Alon, Noga; Tarsi, Michael (1989). "A nowhere-zero point in linear mappings". Combinatorica. 9 (4): 393–395. doi:10.1007/BF02125351. MR 1054015. S2CID 8208350.