Abstract
A particular weighted generalization of some classical zero-sum constants was first considered about 10 years back. Since then, many people got interested in this generalization. Similar generalizations of other zero-sum constants were also considered and these gave rise to several conjectures and questions; some of these conjectures have been established, some of the questions have been answered. And most interestingly, some applications of this weighted generalization have also been found. There are already some expository articles on the classical results as well as on this particular generalization; here we consider the case with a special weight, namely \(\{\pm 1\}\), and dwell mainly on some recent developments not covered in the earlier expository articles.
Dedicated to Prof. Krishna Alladi on the occasion of his 60th birthday
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
S.D. Adhikari, R. Balasubramanian, P. Rath, Some combinatorial group invariants and their generalizations with weights, in Additive Combinatorics. CRM Proceedings and Lecture Notes, vol. 43, ed. by Granville, Nathanson, Solymosi (American Mathematical Society, Providence, RI, 2007), pp. 327–335
S.D. Adhikari, R. Balasubramanian, F. Pappalardi, P. Rath, Some zero-sum constants with weights. Proc. Indian Acad. Sci. (Math. Sci.) 118, 183–188 (2008)
S.D. Adhikari, Y.G. Chen, Davenport constant with weights and some related questions - II. J. Comb. Theory. Ser. A. 115, 178–184 (2008)
S.D. Adhikari, Y.G. Chen, J.B. Friedlander, S.V. Konyagin, F. Pappalardi, Contributions to zero-sum problems. Discret. Math. 306, 1–10 (2006)
S.D. Adhikari, M.N. Chintamani, Number of weighted subsequence sums with weights in \(\{1,-1\}\). Integers 11, paper A 36 (2011)
S.D. Adhikari, D.J. Grynkiewicz, Z.-W. Sun, On weighted zero-sum sequences. Adv. Appl. Math. 48, 506–527 (2012)
S.D. Adhikari, E. Mazumdar, Modifications of some methods in the study of zero-sum constants. Integers 14, paper A 25 (2014)
S.D. Adhikari, P. Rath, Davenport constant with weights and some related questions, in Integers, vol.6, A 30 (2006)
W.R. Alford, A. Granville, C. Pomerance, There are infinitely many Carmichael numbers. Ann. Math. 139(2) no. 3, 703–722 (1994)
N. Alon, M. Dubiner, A lattice point problem and additive number theory. Combinatorica 15, 301–309 (1995)
B. Bollobás, I. Leader, The number of \(k\)-sums modulo \(k\). J. Number Theory 78(1), 27–35 (1999)
M. DeVos, L. Goddyn, B. Mohar, A generalization of Kneser’s addition theorem. Adv. Math. 220, 1531–1548 (2009)
P. Erdős, A. Ginzburg, A. Ziv, Theorem in the additive number theory. Bull. Res. Counc. Isr. 10F, 41–43 (1961)
W.D. Gao, A combinatorial problem on finite abelian groups. J. Number Theory 58, 100–103 (1996)
W.D. Gao, A. Geroldinger, Zero-sum problems in finite abelian groups: a survey. Expo. Math. 24, 337–369 (2006)
W.D. Gao, A. Geroldinger, W.A. Schmid, Inverse zero-sum problems. Acta Arith. 128, 245–279 (2007)
W.D. Gao, R. Thangadurai, A variant of Kemnitz conjecture. J. Comb. Theory. Ser. A. 107, 69–86 (2004)
A. Geroldinger, Additive group theory and non-unique factorizations, in Combinatorial Number Theory and Additive Group Theory, ed. by A. Geroldinger, I. Ruzsa. Advanced Courses in Mathematics - CRM Barcelona, (Birkhäuser Verlag, Basel, 2009)
A. Geroldinger, F. Halter-Koch, Non-unique factorizations: Algebraic, combinatorial and analytic theory. Pure and Applied Mathematics (Boca Raton), 278. (Chapman & Hall/CRC, Boca Raton, 2006)
D.J. Grynkiewicz, Structural Additive Theory, (Springer, Berlin 2013)
D.J. Grynkiewicz, L.E. Marchan, O. Ordaz, A weighted generalization of two theorems of Gao. Ramanujan J. 28(3), 323–340 (2012)
F. Halter-Koch, Arithmetical interpretation of weighted Davenport constants. Arch. Math. (Basel) 103(2), 125–131 (2014)
M. Kneser, Abschätzung der asymptotischen Dichte von Summenmengen. Math. Z. 58, 459–484 (1953)
L.E. Marchan, O. Ordaz, D. Ramos, W.A. Schmid, Some exact values of the Harborth constant and its plus-minus weighted analogue. Arch. Math. (Basel) 101(6), 501–512 (2013)
L.E. Marchan, O. Ordaz, I. Santos, W.A. Schmid, Multi-wise and constrained fully weighted Davenport constants and interactions with coding theory. J. Comb. Theory Ser. A 135, 237–267 (2015)
L.E. Marchan, O. Ordaz, W.A. Schmid, Remarks on the plus-minus weighted Davenport constant. Int. J. Number Theory 10(5), 1219–1239 (2014)
C. Reiher, On Kemnitz’s conjecture concerning lattice points in the plane. Ramanujan J. 13, 333–337 (2007)
K. Rogers, A Combinatorial problem in Abelian groups. Proc. Camb. Phil. Soc. 59, 559–562 (1963)
R. Thangadurai, A variant of Davenport’s constant. Proc. Indian Acad. Sci. (Math. Sci.) 117(2), 147–158 (2007)
H.B. Yu, A simple proof of a theorem of Bollobás and Leader. Proc. Amer. Math. Soc. 131(9), 2639–2640 (2003)
P. Yuan, X. Zeng, A new result on Davenport Constant. J. Number Theory 129, 3026–3028 (2009)
P. Yuan, X. Zeng, Davenport constant with weights. Eur. J. Comb. 31, 677–680 (2010)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Adhikari, S.D. (2017). Plus-Minus Weighted Zero-Sum Constants: A Survey. In: Andrews, G., Garvan, F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series. ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham. https://doi.org/10.1007/978-3-319-68376-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-68376-8_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68375-1
Online ISBN: 978-3-319-68376-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)