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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Inverse theorems for subset sums


Author: Melvyn B. Nathanson
Journal: Trans. Amer. Math. Soc. 347 (1995), 1409-1418
MSC: Primary 11B13; Secondary 11B25, 11B75
DOI: https://doi.org/10.1090/S0002-9947-1995-1273512-1
MathSciNet review: 1273512
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Abstract: Let $A$ be a finite set of integers. For $h \geqslant 1$, let ${S_h}(A)$ denote the set of all sums of $h$ distinct elements of $A$. Let $S(A)$ denote the set of all nonempty sums of distinct elements of $A$. The direct problem for subset sums is to find lower bounds for $|{S_h}(A)|$ and $|S(A)|$ in terms of $|A|$. The inverse problem for subset sums is to determine the structure of the extremal sets $A$ of integers for which $|{S_h}(A)|$ and $|S(A)|$ are minimal. In this paper both the direct and the inverse problem for subset sums are solved.


References [Enhancements On Off] (What's this?)

    G. A. Freiman, On the addition of finite sets. I, Izv. Vyssh. Uchebn. Zaved. Mat. 13 (1959), 202-213. M. B. Nathanson, The simplest inverse problems in additive number theory, Number Theory with an Emphasis on the Markoff Spectrum (A. Pollington and W. Moran, eds.), Marcel Dekker, 1993, pp. 191-206. ---, Additive number theory: $2$ Inverse theorems and the geometry of sumsets, Springer-Verlag, New York, 1995. A. Sárközy, Finite addition theorems. II, J. Number Theory 48 (1994), 197-218.

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Keywords: Additive number theory, subset sums, inverse theorems
Article copyright: © Copyright 1995 American Mathematical Society