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"Circle-squaring."
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Happy pi-day to you!
\"The Cat in the Hat visits Sally and Dick's school on Pi-Day and shows them how to measure pi.\"-- Provided by publisher.
BOOK
Measurable Hall’s theorem for actions of abelian groups
We prove a measurable version of the Hall marriage theorem for actions of finitely generated abelian groups. In particular, it implies that for free measure-preserving actions of such groups and measurable sets which are suitably equidistributed with respect to the action, if they are are equidecomposable, then they are equidecomposable using measurable pieces. The latter generalizes a recent result of Grabowski, Máthé and Pikhurko on the measurable circle squaring and confirms a special case of a conjecture of Gardner.
Journal Article
Borel circle squaring
We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If k ≥ 1 and A, B ⊆ ℝᵏ are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than k, then A and B are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a recent result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the hyperfiniteness of free Borel actions of Zᵈ.
Journal Article
The Circle can be Squared!
Hungarian mathematician Miklos Laczkovich has demonstrated a unique solution to the age-old geometric problem of squaring the circle by breaking a circle up into finite parts and rearranging them as a square.
Journal Article