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Example:
for $n = 2$, $k = 2$, the set $\{ x | -2^2+1\leq x\leq 2^2\}$ can be split into {-1, 1, 2, 4} and {-3, -2, 0, 3}, as $(-1)^2+1^2+2^2+4^2=(-3)^2+(-2)^2+0^2+3^2$

I wonder if this problem has its own name and some related research. I would appreciate it if anyone could give some useful information. Thanks a lot.

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    $\begingroup$ I think it is relevant for you to look at the so called Prouhet Tarry Escott Problem. It is a stronger version than what you ask, namely you need all the first few power sums equal. On the other hand because of the shape of your set in the question note that the Prouhet Thue Morse sequence gives you a solution for the partition into $A,B$ such that the first $k$th powers are equal where $0\leq k\leq N-1$. $\endgroup$
    – Vlad Matei
    Commented Feb 22, 2021 at 13:36

1 Answer 1

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This was solved by Prouhet, M. E. Prouhet. M´emoire sur quelques relations entre les puissances des nombres. C. R. Acad. Sci. Paris. Serie A, 33:31, 1851. A solution (in English) can be found here, page 75.

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