Rational points on the unit sphere

• Autorzy: Eric Schmutz
• Języki publikacji: EN

Abstrakty

EN

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))

Słowa kluczowe

EN

Diophantine approximation; orthogonal group; unitary group; rational points; unit sphere

Kategorie tematyczne

• 14G05: Rational points
• 11J99: None of the above, but in this section

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2008
• Tom: 6
• Numer: 3
• Strony: 482-487

Daty

wydano

2008-09-01

online

2008-07-02

Twórcy

autor

Eric Schmutz

• Drexel University

Bibliografia

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Identyfikatory

DOI

10.2478/s11533-008-0038-4