Zero-sum problem
In number theory, zero-sum problems are certain kinds of combinatorial problems about the structure of a finite abelian group. Concretely, given a finite abelian group G and a positive integer n, one asks for the smallest value of k such that every sequence of elements of G of size k contains n terms that sum to 0. The classic result in this area is the 1961 theorem of Paul Erdős, Abraham Ginzburg, and Abraham Ziv. They proved that for the group Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } of integers modulo n, k = 2 n − 1. {\displaystyle k=2n-1.} Explicitly this says that any multiset of 2n − 1 integers has a subset of size n the sum of whose elements is a multiple of n, but that the same is not true of multisets of size 2n − 2. (Indeed, the lower bound is easy to see: the multiset containing n − 1 copies of 0 and n − 1 copies of 1 contains no n-subset summing to a multiple of n.) This result is known as the Erdős–Ginzburg–Ziv theorem after its discoverers. It may also be deduced from the Cauchy–Davenport theorem. More general results than this theorem exist, such as Olson's theorem, Kemnitz's conjecture (proved by Christian Reiher in 2003), and the weighted EGZ theorem (proved by David J. Grynkiewicz in 2005).
In number theory, zero-sum problems are certain kinds of combinatorial problems about the structure of a finite abelian group. Concretely, given a finite abelian group G and a positive integer n, one asks for the smallest value of k such that every sequence of elements of G of size k contains n terms that sum to 0.
The classic result in this area is the 1961 theorem of Paul Erdős, Abraham Ginzburg, and Abraham Ziv.[1] They proved that for the group of integers modulo n,
Explicitly this says that any multiset of 2n − 1 integers has a subset of size n the sum of whose elements is a multiple of n, but that the same is not true of multisets of size 2n − 2. (Indeed, the lower bound is easy to see: the multiset containing n − 1 copies of 0 and n − 1 copies of 1 contains no n-subset summing to a multiple of n.) This result is known as the Erdős–Ginzburg–Ziv theorem after its discoverers. It may also be deduced from the Cauchy–Davenport theorem.[2]
More general results than this theorem exist, such as Olson's theorem, Kemnitz's conjecture (proved by Christian Reiher in 2003[3]), and the weighted EGZ theorem (proved by David J. Grynkiewicz in 2005[4]).
See also
[edit]References
[edit]- ^ Erdős, Paul; Ginzburg, A.; Ziv, A. (1961). "Theorem in the additive number theory". Bull. Res. Council Israel. 10F: 41–43. Zbl 0063.00009.
- ^ Nathanson (1996) p.48
- ^ Reiher, Christian (2007), "On Kemnitz' conjecture concerning lattice-points in the plane", The Ramanujan Journal, 13 (1–3): 333–337, arXiv:1603.06161, doi:10.1007/s11139-006-0256-y, S2CID 119600313, Zbl 1126.11011.
- ^ Grynkiewicz, D. J. (2006), "A Weighted Erdős-Ginzburg-Ziv Theorem" (PDF), Combinatorica, 26 (4): 445–453, doi:10.1007/s00493-006-0025-y, S2CID 33448594, Zbl 1121.11018.
- Geroldinger, Alfred (2009). "Additive group theory and non-unique factorizations". In Geroldinger, Alfred; Ruzsa, Imre Z. (eds.). 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. pp. 1–86. ISBN 978-3-7643-8961-1. Zbl 1221.20045.
- 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
[edit]- "Erdös-Ginzburg-Ziv theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- PlanetMath Erdős, Ginzburg, Ziv Theorem
- Sun, Zhi-Wei, "Covering Systems, Restricted Sumsets, Zero-sum Problems and their Unification"
Further reading
[edit]- Zero-sum problems - A survey (open-access journal article)
- Zero-Sum Ramsey Theory: Graphs, Sequences and More (workshop homepage)
- Arie Bialostocki, "Zero-sum trees: a survey of results and open problems" N.W. Sauer (ed.) R.E. Woodrow (ed.) B. Sands (ed.), Finite and Infinite Combinatorics in Sets and Logic, Nato ASI Ser., Kluwer Acad. Publ. (1993) pp. 19–29
- Y. Caro, "Zero-sum problems: a survey" Discrete Math., 152 (1996) pp. 93–113
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