Abstract
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:

⊎ ‖rv‖∞ < ε.

⊎ r is also a point on the unit sphere; Σ r _{ i } ^{2} = 1.

⊎ r has rational coordinates; \( r_i = \frac{{a_i }} {{b_i }} \) for some integers a _{ i }, b _{ i }.

⊎ for all \( i,0 \leqslant \left {a_i } \right \leqslant b_i \leqslant (\frac{{32^{1/2} \left\lceil {log_2 n} \right\rceil }} {\varepsilon })^{2\left\lceil {log_2 n} \right\rceil } \) .
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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Schmutz, E. Rational points on the unit sphere. centr.eur.j.math. 6, 482–487 (2008). https://doi.org/10.2478/s1153300800384
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DOI: https://doi.org/10.2478/s1153300800384