正方形剖分成奇数三角形的面积差界
Area difference bounds for dissections of a square into an odd number of triangles
摘要：
1970年的蒙斯基定理指出，一个正方形不能被分割成面积相同的奇数个三角形，但它没有给出面积差的下界。我们将Monsky定理推广到“约束框架映射”，在此基础上，我们可以将半代数几何中的间隙定理应用于多项式面积差测度，从而得到面积差的下界，该下界随三角形个数呈双指数下降.另一方面，我们得到了这个问题的第一个超多项式上界，来自一个明确的建设，使用的Thue-莫尔斯序列。
英文摘要：
Monsky's theorem from 1970 states that a square cannot be dissected into an odd number of triangles of the same area, but it does not give a lower bound for the area differences that must occur. We extend Monsky's theorem to "constrained framed maps"; based on this we can apply a gap theorem from semi-algebraic geometry to a polynomial area difference measure and thus get a lower bound for the area differences that decreases doubly-exponentially with the number of triangles. On the other hand, we obtain the first superpolynomial upper bounds for this problem, derived from an explicit construction that uses the Thue-Morse sequence.
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