Rational points on the unit sphere

Abstract

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; $r_i = \frac{{a_i }} {{b_i }}$ 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))