Logic Seminar

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We show that there exists an equidecomposition between a closed disk and a closed square of the same area in R^2 by translations with algebraic irrational coordinates. Our proof uses a new method for bounding the discrepancy of product sets in the k-torus using only the Erdős–Turán inequality. This resolves a question of Laczkovich from 1990. We also obtain an improved upper bound on the number of pieces required to square the circle, by proving effective bounds on such discrepancy estimates for translations by certain algebraic irrational numbers. This builds on an idea of Frank Calegari for bounding certain sums of products of fractional parts of algebraic numbers, and some computer assistance. This is joint work with Spencer Unger.