Rational points on the sphere

Spatial statistics for lattice points on the sphere I‎: ‎Individual results

‎We study the spatial distribution of point sets on the sphere obtained from the representation of a large integer as a sum of three integer squares‎. ‎We examine several statistics of these point sets‎, ‎such as the electrostatic potential‎, ‎Ripley's function‎, ‎the variance of the number of points in random spherical caps‎, ‎and the covering radius‎. ‎Some of the results are conditional on t...

Rational Points on the Unit Sphere

The unit sphere, centered at the origin in R, has a dense set of points with rational coordinates. We give an elementary proof of this fact that includes explicit bounds on the complexity of the coordinates: for every point v on the unit sphere in R, and every > 0, there is a point r = (r1, r2, . . . , rn) such that: • ||r− v||∞ < . • r is also a point on the unit sphere; P r i = 1. • r has rat...

Distribution of Rational Points: A Survey

Constructing Subdivision Rules from Rational Maps

This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if f is a critically finite rational map with no periodic critical points, then for any sufficiently large integer n the iterate f is the subdivision map of a finite subdivision rule. We are interested here in connections between finite subdivision rules and rational...

Free groups of rotations acting without fixed points on the rational unit sphere

The purpose of this paper is to generalize an example of Sato of a free group of rotations of the Euclidean 3-space whose action on the rational unit sphere is fixed point free. Such groups are of interest, as they can be employed to construct paradoxical decompositions of spheres without assuming the Axiom of Choice.