多边形范数的分离集与Falconer猜想
Separated sets and the Falconer conjecture for polygonal norms
摘要：
Falconer猜想认为，如果E是一个平面集，其豪斯多夫维数严格大于1，则它的欧氏距离集有正的一维勒贝格测度.我们讨论了类似的问题，其中单位球是一个多边形与2K边的非欧几里德范数取代欧几里德距离。证明了对任意范数，存在一个豪斯多夫维数为K/（K-1）的集合，其距离集的勒贝格测度为0.
英文摘要：
The Falconer conjecture asserts that if E is a planar set with Hausdorff dimension strictly greater than 1, then its Euclidean distance set has positive one-dimensional Lebesgue measure. We discuss the analogous question with the Euclidean distance replaced by non-Euclidean norms in which the unit ball is a polygon with 2K sides. We prove that for any such norm there is a set of Hausdorff dimension K/(K-1) whose distance set has Lebesgue measure 0.
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