Rational points on the unit sphere

Eric Schmutz

Open Mathematics (2008)

  • Volume: 6, Issue: 3, page 482-487
  • ISSN: 2391-5455

Abstract

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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 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))

How to cite

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Eric Schmutz. "Rational points on the unit sphere." Open Mathematics 6.3 (2008): 482-487. <http://eudml.org/doc/269530>.

@article{EricSchmutz2008,
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))},
author = {Eric Schmutz},
journal = {Open Mathematics},
keywords = {Diophantine approximation; orthogonal group; unitary group; rational points; unit sphere},
language = {eng},
number = {3},
pages = {482-487},
title = {Rational points on the unit sphere},
url = {http://eudml.org/doc/269530},
volume = {6},
year = {2008},
}

TY - JOUR
AU - Eric Schmutz
TI - Rational points on the unit sphere
JO - Open Mathematics
PY - 2008
VL - 6
IS - 3
SP - 482
EP - 487
AB - 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))
LA - eng
KW - Diophantine approximation; orthogonal group; unitary group; rational points; unit sphere
UR - http://eudml.org/doc/269530
ER -

References

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