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Rational points on the unit sphere

Eric Schmutz — 2008

Open Mathematics

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 = a i b i for some integers a i, b i.⊎ for all i , 0 a i b i ( 32 1 / 2 l o g 2 n ε ) 2 l o g 2 n . One consequence of this...

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