Borel circle squaring
نویسنده
چکیده
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.
منابع مشابه
Squaring the open circle: resolving the iron triangle and the interaction equivalence theorem
The Open University's repository of research publications and other research outputs Squaring the open circle: resolving the iron triangle and the interaction equivalence theorem Conference Item How to cite: Lane, Andrew (2014). Squaring the open circle: resolving the iron triangle and the interaction equivalence theorem. In: OER14 Building communities of open practice , 28-29 April 2014, Newca...
متن کاملSocial Citizenship Rights and the Welfare Circle Dilemma: Attitudinal Findings of Two Chinese Societies
This paper places social citizenship momentum into the context of squaring the welfare circle for examination. Citizenship is a powerful world-level organizing principle especially by the minority groups for their claim of equal treatment. The squaring of welfare circle refers to the need of the governments to constrain their budgets but also meet the rising demands from and needs of their peop...
متن کاملMeasurable circle squaring
Laczkovich proved that if bounded subsets A and B of R have the same non-zero Lebesgue measure and the upper box dimension of the boundary of each set is less than k, then there is a partition of A into finitely many parts that can be translated to form a partition of B. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, thi...
متن کاملTranslation Equidecomposability
Expository piece on M. Laczkovich’s ”Equidecomposability and discrepancy; a solution of Tarski’s circle-squaring problem.” A criterion for translation equidecomposability of two Jordan domains is given with an application.
متن کاملGeometric Constructions and Algebraic Field Extensions
In this paper, we study field extensions obtained by polynomial rings and maximal ideals in order to determine whether solutions exist to three ancient Greek construction problems: squaring the circle, doubling the cube, and trisecting an angle.
متن کامل