Rational points on the unit sphere

By E. Schmutz. 2008

48 citations recorded by the source corpus.

AbstractIt 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 = (r1; r2;…;rn) such that: ⊎ ‖r-v‖∞ < ε.⊎ r is also a point on the unit sphere; Σ ri2 = 1.⊎ r has rational coordinates; $$ r_i = \frac{{a_i }} {{b_i }} $$ for some integers ai, bi.⊎ 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))

Subfields: Computer Graphics and Computer-Aided Design, Computer Vision and Pattern Recognition, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/2de321f469bece234622271ab37e93e8aab8bab2

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In Computer Graphics and Computer-Aided Design

  1. Zhongwei Shen
  2. Jun Geng

In Computer Vision and Pattern Recognition

  1. Yoav Levine
  2. Noam Wies
  3. Amnon Shashua
  4. Zhongwei Shen
  5. Jun Geng
  6. Dacheng Tao
  7. Tamara G. Kolda
  8. Brett W. Bader

In Electrical and Electronic Engineering

  1. Amnon Shashua
  2. Zhongwei Shen
  3. Jun Geng

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