Timeline for Existence of a "generic enough" lattice point interior to a lattice triangle
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Dec 13, 2018 at 2:06 | comment | added | Gerhard Paseman | There are general visibility results by Adikhari which may be a start on this. Essentially, if an integer interval is large enough, it contains an integer coprime to a given integer N, and this has implications to your problem. Search for work of Granville in this area too. Gerhard "Will Help Look For Forest" Paseman, 2018.12.12. | |
| Dec 13, 2018 at 0:26 | history | edited | Avi Steiner | CC BY-SA 4.0 |
added 241 characters in body
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| Dec 13, 2018 at 0:23 | comment | added | Avi Steiner | @fedja hm that would be quite intersecting. I’ll modify the question | |
| Dec 13, 2018 at 0:21 | comment | added | fedja | @AviSteiner If your curiosity stops here, I will, though I believe that life will be more interesting if we modify the question by changing the assumptions to "$T$ contains sufficiently many points in the interior". After all, small size counterexamples in such questions are usually not very exciting or meaningful. | |
| Dec 13, 2018 at 0:13 | comment | added | Avi Steiner | @fedja If you put that as an answer, I'll accept it. | |
| Dec 13, 2018 at 0:09 | comment | added | fedja | How about (0,0), (3,2),(1,3) for the vertices of $T$? | |
| Dec 12, 2018 at 21:22 | comment | added | Avi Steiner | @MarcusM I didn't even think about that! I've made a clarification of what I mean. I also clarified that I want p to be in the interior of T. | |
| Dec 12, 2018 at 21:21 | history | edited | Avi Steiner | CC BY-SA 4.0 |
clarified the definition of T and added that I want p to be in the interior
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| Dec 12, 2018 at 21:18 | comment | added | Marcus M | When you say $p \in T \cap \mathbb{Z}^2$, it looks like you mean for $T$ to include the interior of the triangle; however, your condition for $p$ that you seek is more ambiguous. It says $\text{conv}(p,v) = \{p,v\}$ for all $v \in T$, however in your image you only tested in for $v$ being the extreme points (i.e. the three vertices) of the triangle. Is that the condition that you seek? | |
| Dec 12, 2018 at 21:16 | comment | added | Avi Steiner | @MarcusM I'm not sure what you mean. | |
| Dec 12, 2018 at 21:15 | comment | added | Marcus M | It looks to me like your definition of $T$ is a bit ambiguous, as it first refers to the entire triangle (with interior), but later (possibly) refers to only the three vertices of it. | |
| Dec 12, 2018 at 20:39 | history | asked | Avi Steiner | CC BY-SA 4.0 |