Journal article
Rational points on the unit sphere
Open mathematics (Warsaw, Poland), v 6(3), pp 482-487
01 Sep 2008
Featured in Collection : UN Sustainable Development Goals @ Drexel
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))
Metrics
Details
- Title
- Rational points on the unit sphere
- Creators
- Eric Schmutz - Drexel University
- Publication Details
- Open mathematics (Warsaw, Poland), v 6(3), pp 482-487
- Publisher
- Versita
- Number of pages
- 6
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000258624500013
- Scopus ID
- 2-s2.0-47349120711
- Other Identifier
- 991019168778404721
UN Sustainable Development Goals (SDGs)
This publication has contributed to the advancement of the following goals:
InCites Highlights
Data related to this publication, from InCites Benchmarking & Analytics tool:
- Web of Science research areas
- Mathematics