# Rational points on the unit sphere

Open Mathematics (2008)

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

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topEric 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 -

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