Non-Euclidean visibility problems
نویسندگان
چکیده
Consider in R2 the standard lattice L = Z×Z and the origin (0,0). A point (m,n) ∈ L−{(0,0)} is said to be visible if the segment connecting the origin and (m,n) does not contain any other lattice points. Visibility problems have been studied since a century. Perhaps the most celebrated problems are the visible version of Gauss circle problem and the so-called orchard problem (see other problems in [5]). In both of these problems one considers visible lattice points in a large circle. The first problem consists of approximating the cardinality of this set of points. It turns out that improvements on the trivial bounds of the error term are related to Riemann Hypothesis (see [12]). In the orchard problem the visible points are considered to be thick and it is asked the minimal thickness such that all exterior points are eclipsed. In the formulation included in p. 150 of [14], “How thick must the trunks of the trees in a regularly spaced circular orchard grow if they are to block completely the view from the center?”. In contrast with the previous problem, orchard problem can be considered as solved in a wide sense (see [2]) by elementary methods. In this paper we deal with the hyperbolic analog of visibility problems. Namely, we consider Poincaré’s plane H, i.e., the upper half plane endowed with the metric
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