Abstract
Let A 1,..., A r be finite, nonempty sets of integers, and let h 1,..., h r be positive integers. The linear form h 1 A 1 +...+ h r A r is the set of all integers of the form b 1+...+b r , where b i is an integer that can be represented as the sum of h i elements of the set A,. In this paper, the structure of the linear form h 1 A 1+...+h r A r is completely determined for all sufficiently large integers h i .
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
Preview
Unable to display preview. Download preview PDF.
References
A.G. Khovanskii, “Newton polyhedron, Hilbert polynomial, and sums of finite sets” Funktsional Anal i Prilozhen. 26 (1992), 276–281.
A.G. Khovanskii, “Sums of finite sets, orbits of commutative semigroups, and Hilbert functions,” Funktsional. Anal i Prilozhen. 29 (1995), 102–112.
V.F. Lev, “Structure theorem for multiple addition and the Frobenius problem,” J. Number Theory 58 (1996), 79–88.
V.F. Lev, “Addendum to ‘Structure theorem for multiple addition’,” J. Number Theory 65 (1997), 96–100.
M.B. Nathanson, “Sums of finite sets of integers,” Amer. Math. Monthly 79 (1972), 1010–1012.
M.B. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets, volume 165 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1996.
M.B. Nathanson, “Growth of sumsets in abelian semigroups,” Preprint, 1997.
Ö.J. Rödseth, “On h-bases for n,” Math. Scand. 48 (1981), 165–183.
Author information
Authors and Affiliations
Editor information
Additional information
Dedicated to Paul Erdős
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Han, SP., Kirfel, C., Nathanson, M.B. (1998). Linear Forms in Finite Sets of Integers. In: Alladi, K., Elliott, P.D.T.A., Granville, A., Tenebaum, G. (eds) Analytic and Elementary Number Theory. Developments in Mathematics, vol 1. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-4507-8_16
Download citation
DOI: https://doi.org/10.1007/978-1-4757-4507-8_16
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-5058-1
Online ISBN: 978-1-4757-4507-8
eBook Packages: Springer Book Archive