Problem #494: If $A\subset \mathbb{C}$ is a finite set and $k\geq 1$ then...

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Progress

A problem of Selfridge and Straus [SeSt58], who prove that this is true if $k=2$ and $\lvert A\rvert \neq 2^l$ (for $l\geq 0$). On the other hand, there are examples with two distinct $A,B$ both of size $2^l$ such that $A_2=B_2$.

Selfridge and Straus [SeSt58] also show that the answer is yes if $k=3$ and $\lvert A\rvert>6$ or $k=4$ and $\lvert A\rvert>12$.

More generally, they prove that $A$ is uniquely determined by $A_k$ if $\lvert A\rvert$ is divisible by a prime greater than $k$.

Kruyt notes that this is trivially false if $\lvert A\rvert=k$ then rotating $A$ around an appropriate point produces a distinct set with the same sum. Presumably some condition like '$\lvert A\rvert$ sufficiently large' is intended. Tao also notes that this is false if $\lvert A\rvert=2k$.

Gordon, Fraenkel, and Straus [GFS62] proved that, for all $k$, the multiset $A_k$ uniquely determines $A$ provided $\lvert A\rvert$ is sufficiently large.

(In [Er61] Erdős states this problem incorrectly, replacing sums with products. This product formulation is easily seen to be false, as observed by Steinerberger: consider the case $k=3$ and subsets of the 6th roots of unity corresponding to $\{0,1,2,4\}$ and $\{0,2,3,4\}$ (as subsets of $\mathbb{Z}/6\mathbb{Z}$). The correct problem statement can be found in the paper of Selfridge and Straus that Erdős cites.)

This is mentioned in problem C5 of Guy's collection [Gu04].