Abstract
Let A_1, ..., A_r be finite, nonempty sets of integers, and let h_1,..., h_r be positive integers. The linear formh_1A_1 + ··· + h_rA_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 ofh_i elements of the set A_i. In this paper, the structure of the linear form h_1A_1 + ··· + h_rA_r is completely determined for all sufficiently large integersh_i .
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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.
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Han, SP., Kirfel, C. & Nathanson, M.B. Linear Forms in Finite Sets of Integers. The Ramanujan Journal 2, 271–281 (1998). https://doi.org/10.1023/A:1009734613675
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DOI: https://doi.org/10.1023/A:1009734613675