Rational points on the unit sphere
Eric Schmutz · Open Mathematics · 2008
It is known that the unit sphere, centered at the origin in ℝ n , 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 ℝ n , and every ν > 0; there is a point r = (r 1; r 2;…;r n) such that: 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))