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A non-Borel union of unit half-open squares

On the complex plane $\mathbb C$ consider the half-open square $$\square=\{z\in\mathbb C:0\le\Re(z)<1,\;0\le\Im(z)<1\}.$$ Observe that for every $z\in \mathbb C$ and $p\in\{0,1,2,3\}$ the set $...