[arutar]

I'm not sure why this is not mentioned in the article, but there is nothing special about circles and squares (or 2 dimensions, for that matter). If anything, phrasing it like this gives the (misleading) impression that somehow features of squares and circles are important!

The authors proved [1, Thm. 1.3] that given any two sets in R^d with equal non-zero measure and boundaries that are "not too horrible" (i.e. box / Minkowski dimensions of their boundaries less than d), one can cut one of the sets into finitely many Borel pieces and rearrange them (i.e. apply isometries in R^d) to obtain the other set.

You can also guarantee that the pieces have positive measure under a mild technical assumption.

[1] https://arxiv.org/pdf/2202.01412.pdf