Uniformly Discrete Forests with Poor Visibility. (9th October 2017) - ALON, NOGA
Abstract : We prove that there is a set F in the plane so that the distance between any two points of F is at least 1, and for any positive ϵ < 1, and every line segment in the plane of length at least ϵ −1− o (1), there is a point of F within distance ϵ of the segment. This is tight up to the o (1)-term in the exponent, improving earlier estimates of Peres, of Solomon and Weiss, and of Adiceam.