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R. Guy, Unsolved problems in number theory, 3rd edition, Springer, 2004.

In this book, on page 167-168, Problem C5, Sums determining members of a set, discusses a question Leo Moser asked: suppose $X\subset\mathbb{Z}$ is an $n$-element set and $A$ be set of all $k$-element sums of subsets of $X$. Is there any other $Y\neq X$ such that $Y\subset\mathbb{Z}$ such that its $k$-element sums are exactly $A$?

Only small cases were known: $k=2,3,4,5$ and any $n$.

Question. What is the general status of this problem for $k>5$? Max Alekseyev provided some references.

Update. I'm still interested in the question: are there (non-cosmetic) equivalent formulations of the above problem?

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In a recent paper A set of 12 numbers is not determined by its set of 4-sums (in Russian), Isomurodov and Kokhas identify an error in the argument of Ewell (1968) and present two distinct sets of 12 integers with the same multiset of 4-sums.

They also give a nice overview of the area and provide useful references. For the general case $(n,k)$, they mention the following results:

  • Selfridge and Straus (1958) describe a union of Diophantine equations such that for $(n,k)$ not satisfying any of the equations (a typical situation), the set $X$ is unique. For $(n,k)$ satisfying at least one of the equations, the uniqueness of $X$ remains an open question (the case $(12,4)$ was such).
  • Fraenkel, Gordon, and Straus (1962) prove that for each $k$, there exists only a finite number of values of $n$, for which $X$ may be not unique.
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