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Claim. The answer to Problem 1124 is yes. Miklós Laczkovich, Equidecomposability and discrepancy; a solution of Tarski's circle-squaring problem, J. Reine Angew. Math. 404 (1990), 77--117, proves that a closed disk and a square of the same area can be partitioned into the same finite number of pieces so that the pieces of the two partitions are pairwise congruent; the congruences can all be taken to be translations, which is more than the question asks. The proof is nonconstructive: it uses the axiom of choice, and the pieces need not be Lebesgue measurable. Its method combines discrepancy estimates for the distribution of lattice translates with a combinatorial matching argument. The general theorem that two bounded sets in $\mathbb R^k$ of the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than $k$ are equidecomposable by translations is Laczkovich's later paper, Decomposition of sets with small boundary, J. London Math. Soc. (2) 46 (1992), 58--64, as Marks and Unger cite it; the 1990 paper settles the disk and the square. No file of either paper is held; the statement follows the site's page and the account in A. Marks and S. Unger, Borel circle squaring, Ann. of Math. (2) 186 (2017), no. 2, 581--605 (arXiv:1612.05833), whose abstract describes their result as a Borel version of Laczkovich's theorem.

Depends on. No page of this wiki.

Acceptance. The result is refereed: it appeared in the Journal für die reine und angewandte Mathematik. The site's curator, Thomas Bloom, marks the problem proved and credits Laczkovich, recording that the decomposition uses translations only (the site's page, prints no last-edited date). Two later theorems strengthen the result for the same class of sets without changing the answer: Grabowski, Máthé and Pikhurko (Ann. of Math. (2) 185 (2017), no. 2, 671--710; arXiv:1501.06122) make the pieces Lebesgue and Baire measurable, and Marks and Unger make them Borel, answering a question of Wagon; the site's thread records Laczkovich's general theorem in a comment, citing Marks and Unger, and neither is paged.