It is known that the unit sphere, centered at the origin in ℝn, 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 ν on the unit sphere in ℝn, and every ν > 0; there is a point r = (r 1; r 2;…;r n) such that: ⊎ ‖r-v‖∞ < ε.⊎ r is also a point on the unit sphere; Σ r i 2 = 1.⊎ r has rational coordinates; for some integers a i, b i.⊎ for all . 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))
@article{bwmeta1.element.doi-10_2478_s11533-008-0038-4,
author = {Eric Schmutz},
title = {Rational points on the unit sphere},
journal = {Open Mathematics},
volume = {6},
year = {2008},
pages = {482-487},
zbl = {1176.11037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0038-4}
}
Eric Schmutz. Rational points on the unit sphere. Open Mathematics, Tome 6 (2008) pp. 482-487. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0038-4/
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