Inverse theorems for subset sums
Author:
Melvyn B. Nathanson
Journal:
Trans. Amer. Math. Soc. 347 (1995), 14091418
MSC:
Primary 11B13; Secondary 11B25, 11B75
DOI:
https://doi.org/10.1090/S00029947199512735121
MathSciNet review:
1273512
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
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.

G. A. Freiman, On the addition of finite sets. I, Izv. Vyssh. Uchebn. Zaved. Mat. 13 (1959), 202213.
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. 191206.
, Additive number theory: $2$ Inverse theorems and the geometry of sumsets, SpringerVerlag, New York, 1995.
A. Sárközy, Finite addition theorems. II, J. Number Theory 48 (1994), 197218.
Retrieve articles in Transactions of the American Mathematical Society with MSC: 11B13, 11B25, 11B75
Retrieve articles in all journals with MSC: 11B13, 11B25, 11B75
Additional Information
Keywords:
Additive number theory,
subset sums,
inverse theorems
Article copyright:
© Copyright 1995
American Mathematical Society