Rational Points on the Unit Sphere
نویسنده
چکیده
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 rational coordinates; ri = ai bi for some integers ai, bi. • for all i, 0 ≤ |ai| ≤ bi ≤ ( 32 dlog2 ne )2dlog2 ne. One consequence of this result is a relatively simple and quantitative proof of the fact that the rational orthogonal group O(n,Q) is dense in O(n,R) with the topology induced by Frobenius’ matrix norm. Unitary matrices in U(n,C) can likewise be approximated by matrices in U(n,Q(i)).
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تاریخ انتشار 2008