Timeline for A point set of power series with coefficients in {-1, 1}. Connected or not?
Current License: CC BY-SA 3.0
21 events
| when toggle format | what | by | license | comment | |
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| Nov 21, 2015 at 12:20 | comment | added | Todd Trimble | Where does this question come from? What's the context/motivation? | |
| Nov 21, 2015 at 0:52 | history | edited | YCor | CC BY-SA 3.0 |
changed tags; slightly rephrased the question since the set $M$ cannot be described in a naive way
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| S Nov 20, 2015 at 21:11 | history | suggested | Nikita Sidorov |
just an extra tag
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| Nov 20, 2015 at 20:51 | comment | added | YCor | It's very easy to check that for $|z|=1/2$ then $z\in M$ (the set of $z$ st $X_z$ is connected) iff $z$ is real. I think it's known to hold when $|z|=1/2+\epsilon$ for small enough $\epsilon$ but it sounds much harder. Also an exercise: $X_z$ is either a Cantor set or is connected. Also there's a simple argument based on Hausdorff dimension showing that if $|z|\ge 1-\epsilon'$ then $z\in M$. | |
| Nov 20, 2015 at 20:45 | review | Suggested edits | |||
| S Nov 20, 2015 at 21:11 | |||||
| Nov 20, 2015 at 18:27 | answer | added | Nikita Sidorov | timeline score: 7 | |
| Nov 20, 2015 at 12:10 | vote | accept | Kirby Lee | ||
| Nov 20, 2015 at 12:03 | answer | added | Douglas Zare | timeline score: 23 | |
| Nov 20, 2015 at 10:57 | history | edited | Douglas Zare | CC BY-SA 3.0 |
Clarified question. Added dynamical systems tag.
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| Nov 20, 2015 at 10:10 | comment | added | Sebastian Goette | @user281042 Sorry, I misunderstood the question the way it was phrased before the edit. I deleted the comment above. | |
| Nov 20, 2015 at 9:21 | comment | added | Benoît Kloeckner | @IlyaBogdanov: you are right, but even the $|z|=1/2$ case is interesting. If $z=\pm1/2$ then $X_z$ is a segment, but I guess it is a Cantor set in every other case. Certainly, when $|z|>1/2$ there are more possibilities. | |
| Nov 20, 2015 at 9:18 | answer | added | Benoît Kloeckner | timeline score: 13 | |
| Nov 20, 2015 at 9:17 | comment | added | Ilya Bogdanov | Take $|z|=1/3$. If $a_1=1$, then the corresponding element of $X_z$ is at distance of $\leq 1/6$ from $z$, otherwise it is at distance of $\leq 1/6$ from $-z$. Thus $X_z$ is disconnected. A more interesting question arises when $|z|>1/2$ (and now it may depend on the argument of $z$ as well)... | |
| Nov 20, 2015 at 9:03 | comment | added | Kirby Lee | @BenoîtKloeckner Thanks a lot. It's exactly the question. I'm a little ashamed for my poor description... | |
| Nov 20, 2015 at 8:57 | comment | added | Benoît Kloeckner | I took the liberty of rephrasing your question, let me know if this is not what you wanted to ask. I think the question is nice and deserves an attention it would not have gotten in your phrasing. | |
| Nov 20, 2015 at 8:56 | history | edited | Benoît Kloeckner | CC BY-SA 3.0 |
Rephrased the question more clearly
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| Nov 20, 2015 at 8:56 | review | Close votes | |||
| Nov 20, 2015 at 13:43 | |||||
| Nov 20, 2015 at 8:53 | history | edited | Kirby Lee | CC BY-SA 3.0 |
edited tags; edited title
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| Nov 20, 2015 at 8:52 | comment | added | Kirby Lee | @SebastianGoette Thx for your comment, but I think I had wronged the idea... For a fixed z it's not a power series. And the point is determined by the signature of $\{a_n\}$. Would you please give me more information if you know some details about this question? | |
| Nov 20, 2015 at 8:35 | review | First posts | |||
| Nov 20, 2015 at 8:38 | |||||
| Nov 20, 2015 at 8:35 | history | asked | Kirby Lee | CC BY-SA 3.0 |