Timeline for Continued fractions using all natural integers
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 17, 2014 at 13:51 | comment | added | XL _At_Here_There | It has measure 0 and is uncountable? | |
| May 8, 2010 at 6:35 | comment | added | François G. Dorais | Regarding the last sentence. The space is homeomorphic to the the irrationals. It's rather unlike the Cantor set since its compact subsets are nowhere dense. | |
| May 7, 2010 at 14:19 | comment | added | Wadim Zudilin | @Gerald: Although I've not seen the conjecture about transcendence of such CFs (but this is indeed a very unnatural set as already pointed out), it's something that one should expect. The CF for $[1,2,3,\dots]$ (as quotient of the values of Bessel functions) is trancendental. @David: almost all numbers have approximation rate consistent with algebraic numbers. Algebraic numbers are too rare to appear. | |
| May 7, 2010 at 11:48 | comment | added | David E Speyer | I should point out that my answer only argues that the approximation rate of this continued fraction is consistent with the set containing an algebraic number; I have no opinion as to whether there actually is an algebraic number in this set. | |
| May 6, 2010 at 21:46 | comment | added | Steven Gubkin | @Prof. Edgar: See the answers of Timothy Gowers and David Speyer below . | |
| May 6, 2010 at 17:59 | comment | added | Gerald Edgar | Is there any algebraic number at all in this set? Continued fraction [1,2,3,4,5...] itself evaluates to a quotient of Bessel functions, so is presumably not algebraic. | |
| Dec 16, 2009 at 13:41 | vote | accept | timur | ||
| Nov 20, 2009 at 8:04 | comment | added | Harrison Brown | Actually, I think it's nowhere dense. Topologically this resembles the Cantor set, except I'm not sure if it's dense in itself or not. | |
| Nov 20, 2009 at 6:28 | history | answered | Harrison Brown | CC BY-SA 2.5 |