Timeline for Does there exist a real-analytic curve in the plane that is injective and dense?
Current License: CC BY-SA 4.0
Post Revisions
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 1, 2025 at 5:05 | vote | accept | Daniel Asimov | ||
| Jan 4, 2025 at 22:18 | answer | added | coudy | timeline score: 17 | |
| Jan 4, 2025 at 5:49 | answer | added | Moishe Kohan | timeline score: 15 | |
| Jan 3, 2025 at 23:23 | comment | added | fedja | @KevinCasto Distant points are controlled by the small error of the approximation to a continuous curve you draw adding piece by piece already. The derivative is needed only to avoid self-intersections in a very near vicinity (if you have a non-self intersecting continuous $f$, its uniform approximation $g$ certainly won't glue faraway parameters, but can glue very close ones as much as it wants unless you control something like the derivative as well). | |
| Jan 3, 2025 at 22:10 | comment | added | Kevin Casto | @fedja Can you say a bit more about how controlling/approximating the derivative allows you to prevent self-intersections between distant input points? | |
| Jan 3, 2025 at 15:38 | comment | added | Alexandre Eremenko | This is certainly so if the curve is parametrized by a non-compact interval. But if the curve is parametrized by $[a,b]$ the answer is negative. | |
| Jan 3, 2025 at 15:23 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
Added missing space
|
| Jan 3, 2025 at 2:30 | history | edited | Malik Younsi |
edited tags
|
|
| Jan 3, 2025 at 2:00 | comment | added | fedja | Yes. A standard theorem (you can find it in most textbooks on approximation theory) is that for every positive continuous function $\delta$ on $\mathbb R$ and any continuous $f$ there, one can find a real analytic (even entire) $g$ such that $|g-f|\le\delta$ on the line. This doesn't yet give you what you want if used just as a black box, but if you look at the proof, you'll be able to modify it pretty easily to answer your question. To get 1-1, it helps to approximate the derivative together with $f$ itself. If you still have trouble with it, let me know and I'll post more details :-) | |
| Jan 3, 2025 at 1:39 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
injective and dense —> both injective and dense
|
| Jan 3, 2025 at 1:06 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
Corrected interval [1/2, 1) —> (1/2, 1].
|
| Jan 3, 2025 at 1:00 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
Corrected interval [1/2, 1) —> (1/2, 1].
|
| Jan 2, 2025 at 23:48 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
Rearranged and fine-tuned the question and added a reference
|
| Jan 2, 2025 at 17:48 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
Improved wording
|
| Jan 2, 2025 at 17:43 | history | asked | Daniel Asimov | CC BY-SA 4.0 |