Timeline for A naive diophantine approximation question
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| S Jul 7, 2017 at 19:59 | history | suggested | Luc Guyot | CC BY-SA 3.0 |
If $n$ can be zero, the answer is trivially "yes"
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| Jul 7, 2017 at 19:44 | review | Suggested edits | |||
| S Jul 7, 2017 at 19:59 | |||||
| Jul 7, 2017 at 10:46 | comment | added | Yaakov Baruch | @GerryMyerson: ah - the key is whether the word "always" in the question refers to $\epsilon$, or to both $\alpha$ and $\epsilon$... | |
| Jul 5, 2017 at 22:50 | comment | added | Gerry Myerson | Yaakov says the answer is no, Luc says the answer is yes, and they're both right. | |
| Jul 5, 2017 at 20:03 | answer | added | Yaakov Baruch | timeline score: 11 | |
| Jul 5, 2017 at 17:30 | answer | added | Luc Guyot | timeline score: 5 | |
| Jul 5, 2017 at 5:23 | comment | added | Gerry Myerson | Yes, Igor, but any nontrivial information about the fractional part of $(3/2)^n$ seems to be hard to come by. Mahler proved $\|(3/2)^k\|>(3/4)^k$ for $k$ sufficiently large, where $\|x\|$ is the distance from $x$ to the nearest integer (I realize this doesn't speak to your question, it's just an example of the kind of thing that is known). Habsieger, Acta Arith. 106 (2003), no. 3, 299–309, has $\|(3/2)^k\|>.57434^k$ for $k\ge5$. | |
| Jul 5, 2017 at 4:41 | comment | added | Noam D. Elkies | . . . and if it's a Salem number then $\|\alpha^n\|$ has a known distribution and does get arbitrarily (albeit not exponentially) small. en.wikipedia.org/wiki/Salem_number | |
| Jul 5, 2017 at 1:25 | comment | added | Robert Israel | Slightly more generally than for the golden ratio, if $\alpha$ is a Pisot-Vijayaraghavan number, $\|\alpha^n \| = \inf_{m \in \mathbb N} |\alpha^n - m| \to 0$ exponentially as $n \to \infty$. | |
| Jul 5, 2017 at 0:36 | comment | added | Igor Rivin | @GerryMyerson Some would argue that the question I ask is somewhat easier than the question of distribution. | |
| Jul 4, 2017 at 23:34 | comment | added | Gerry Myerson | @René, the question of the distribution of the fractional parts of powers of $3/2$ is a notorious open problem, related to Waring's problem. See, e.g., en.wikipedia.org/wiki/Waring%27s_problem | |
| Jul 4, 2017 at 23:09 | comment | added | R.P. | I have trouble answering this even for $\alpha=3/2$... | |
| Jul 4, 2017 at 22:03 | comment | added | Noam D. Elkies | True for the golden ratio, at any rate. But that's unusual because for large $n$ every $\alpha^n$ is close to an integer. | |
| S Jul 4, 2017 at 21:52 | history | suggested | jeq | CC BY-SA 3.0 |
Removed stray opening parenthesis.
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| Jul 4, 2017 at 21:51 | review | Suggested edits | |||
| S Jul 4, 2017 at 21:52 | |||||
| Jul 4, 2017 at 21:28 | history | asked | Igor Rivin | CC BY-SA 3.0 |