Timeline for Is $\pi$ well-approximable?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 11, 2012 at 15:40 | comment | added | Lee Mosher | mathworld.wolfram.com/PiContinuedFraction.html gives some information about this question, which seems to confirm that this is open. | |
| Jun 11, 2012 at 15:32 | comment | added | James Worrell | @Emil: Yes, that is an equivalent formulation. Are the partial convergents in the cf of $\pi$ unbounded? | |
| Jun 11, 2012 at 15:24 | comment | added | Kevin O'Bryant | @Emil: you're right; I missed the "for all c" clause. | |
| Jun 11, 2012 at 15:16 | comment | added | George Lowther | Using the standard pigeonhole argument to show that there are infinitely many solutions for some $c$, the question would seem to reduce to the statement that the minimum distance between the first $n$ multiples of $\pi$ taken modulo 1 goes to zero faster than 1/n. That is, they are not too close to equally spaced. Certainly holds true for almost all real numbers, but I'm not aware of any theorem saying that it is true for $\pi$. | |
| Jun 11, 2012 at 15:05 | comment | added | Emil Jeřábek | Note that the question asks for all $c>0$, not for some $c>0$. | |
| Jun 11, 2012 at 14:59 | comment | added | Emil Jeřábek | @Kevin: It may be just my ignorance in the subject, but it would seem to me that the statement the OP wants would require the terms of the cf of $\pi$ to be unbounded, and this seems to be an open problem. | |
| Jun 11, 2012 at 14:49 | comment | added | Kevin O'Bryant | This is a basic result from continued fractions. Any intro number theory text that has a section on cf's will contain this result. | |
| Jun 11, 2012 at 14:38 | comment | added | Alex R. | yes: mathoverflow.net/questions/53724/… | |
| Jun 11, 2012 at 13:58 | history | asked | James Worrell | CC BY-SA 3.0 |