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

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 $(n,k)=(12,4)$ was such).
  • Fraenkel, Gordon, and Straus (1962) prove that for each $k$, there exists only a finite number of values of $n$ such that $X$ is not unique.

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.
Source Link
Max Alekseyev
  • 34.3k
  • 5
  • 74
  • 152

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 $(n,k)=(12,4)$ was such).
  • Fraenkel, Gordon, and Straus (1962) prove that for each $k$, there exists only a finite number of values of $n$ such that $X$ is not unique.