Let $A\subset \mathbb{C}$ be a finite set, for any $k\geq 1$ let
\[A_k = \{ z_1\cdots z_k : z_i\in A\textrm{ distinct}\}.\]
For $k>2$ does the set $A_k$ uniquely determine the set $A$?

A problem of Selfridge and Straus [SeSt58], who prove that this is true if $k=2$ and $\lvert A\rvert \neq 2^l$ (for $k\geq 0$).