On Sets Which Meet Each Line in Exactly Two Points
نویسنده
چکیده
ABSTRACT Using techniques from geometric measure theory and descriptive set theory we prove a general result concerning sets in the plane which meet each straight line in exactly two points As an application we show that no such two point set can be expressed as the union of countably many recti able sets together with a set with Hausdor measure zero Also as another corollary we show that no analytic set can be a two point set provided every purely unrecti able set meets some line in at least three points Some generalizations are given to n point sets and some other geometric constructionsUsing techniques from geometric measure theory and descriptive set theory we prove a general result concerning sets in the plane which meet each straight line in exactly two points As an application we show that no such two point set can be expressed as the union of countably many recti able sets together with a set with Hausdor measure zero Also as another corollary we show that no analytic set can be a two point set provided every purely unrecti able set meets some line in at least three points Some generalizations are given to n point sets and some other geometric constructions To the memory of Paul Erd os
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