Rational points on the unit sphere |
| |
Authors: | Eric Schmutz |
| |
Institution: | (1) Mathematics Department, Drexel University, Philadelphia, Pennsylvania, 19104, USA |
| |
Abstract: | It is known that the unit sphere, centered at the origin in ℝ
n
, 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 ℝ
n
, 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;
for some integers a
i
, b
i
.
|
– |
⊎ for all
.
|
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))
|
| |
Keywords: | Diophantine approximation orthogonal group unitary group rational points unit sphere |
本文献已被 SpringerLink 等数据库收录! |
|