Timeline for Continued fractions using all natural integers
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| May 7, 2010 at 15:05 | comment | added | Wadim Zudilin | There is nothing to do with Roth's theorem! The number $[0,1,2,\dots]$ has irrationality exponent 2 (as algebraics) and even the exact quality of its approximation by rationals is known (see B.\,G.~Tasoev's paper [Math. Notes 67 (2000), 786--791]). There are no such results for algebraic irrationalities of degree $>2$. | |
| May 7, 2010 at 14:30 | history | edited | Sidney Raffer | CC BY-SA 2.5 |
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| May 7, 2010 at 13:54 | comment | added | Wadim Zudilin | A classical (Lambert-Euler) continued fraction is $(e-1)/(e+1)=[0;2,6,10,14,\dots]$ (Lang's or Cassels's books on Diophantine approximations). In a similar way one can show that $[1,2,3,\dots]$ is of the form $I_0(2)/I_0(1)$ (or something like this) for Bessel's functions. The algebraic independence of the numerator and denominator in this quotient was shown by C. Siegel already in 1929. | |
| May 7, 2010 at 12:12 | comment | added | Sidney Raffer | Oh!! Do you have a reference? Are you sure you're not thinking of Champernowne's constant .1234.... ? | |
| May 7, 2010 at 11:46 | comment | added | Wadim Zudilin | But $[0,1,2,3,\dots]$ is known to be a transcendental number! | |
| May 7, 2010 at 10:24 | history | edited | Sidney Raffer | CC BY-SA 2.5 |
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| May 7, 2010 at 9:31 | history | answered | Sidney Raffer | CC BY-SA 2.5 |