Timeline for Bounded differences in exponential sequences
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Dec 15, 2013 at 1:32 | comment | added | Greg Martin | I don't know for sure whether this will work; as you point out, whether or not there are powers of $2$ in the right intervals isn't clear until you try it. As for a closed form, I doubt this method has any hope of that. | |
| Dec 14, 2013 at 23:18 | comment | added | Dylan Pizzo | @GregMartin Thanks. I apologize if this is a stupid question, but how do you know that there will be a power of 2 in a power of that interval? Other than that, it seems like this should work. Also, do you know of any approach I could use to attempt to get a closed form for the limit? | |
| Dec 14, 2013 at 20:05 | comment | added | Greg Martin | @Dpiz: excellent point! As stated, that is a real danger. But there are only countably many rational powers of $2$; so after the $n$th recursive step, we could diminish the interval if necessary to make sure it doesn't include the $n$th rational power of $2$ (in some fixed enumeration). | |
| Dec 14, 2013 at 13:28 | comment | added | Dylan Pizzo | @GregMartin Why would this method not simply converge to one of the rational powers of two in that inteval? | |
| Dec 13, 2013 at 21:47 | comment | added | Greg Martin | I suspect it should be possible to construct two real numbers that are nontrivially close, recursively: for simplicity fix one of them to equal $2$. Given a tiny interval such that any real number in that interval has $m$ powers that are within $k$ of powers of $2$, look far far out until a large power of that interval contains another power of $2$, and then reduce the tiny interval even further so that the large power of the interval is all within $k$ of the power of $2$. Now we have $m+1$ such close calls, and recurse. | |
| Dec 13, 2013 at 21:40 | comment | added | Dylan Pizzo | Yes. Thanks. I edited the question. | |
| Dec 13, 2013 at 21:39 | history | edited | Dylan Pizzo | CC BY-SA 3.0 |
changed definiton of trivially close
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| Dec 13, 2013 at 21:30 | comment | added | Gerry Myerson | I think your definition of "trivially close" should be extended to cover the case where one of the numbers is a (positive) rational power of the other, e.g., $\sqrt2$ and $\root3\of2$. | |
| Dec 13, 2013 at 21:21 | vote | accept | Dylan Pizzo | ||
| Dec 13, 2013 at 21:17 | answer | added | Joe Silverman | timeline score: 3 | |
| Dec 13, 2013 at 20:55 | history | edited | Dylan Pizzo | CC BY-SA 3.0 |
Changed title from series to more accurate sequences
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| Dec 13, 2013 at 20:53 | review | First posts | |||
| Dec 13, 2013 at 21:01 | |||||
| Dec 13, 2013 at 20:36 | history | asked | Dylan Pizzo | CC BY-SA 3.0 |