### Rational points on the unit sphere

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 $$i,0\u2a7d\left|{a}_{i}\right|\u2a7d{b}_{i}\u2a7d{\left(\frac{{32}^{1/2}\u2308lo{g}_{2}n\u2309}{\epsilon}\right)}^{2\u2308lo{g}_{2}n\u2309}$$ . One consequence of this...