On the sum of squares of middle-third Cantor set

Let $C$ be the middle-third Cantor set. In this paper, we show that for every $x\in [0,4]$, there exist $x_1, x_2, x_3, x_4 \in C$ such that $x= x_1^2+x_2^2+x_3^2+x_4^2,$ which answers a question posed by Athreya, Reznick,and Tyson.

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