Timeline for Largest set of integers without 3-term arithmetic progressions mod $n$
Current License: CC BY-SA 3.0
10 events
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| Jan 4, 2013 at 2:09 | comment | added | Robert Israel | Re: "more irregular": note in particular that $e_3(Z_n)$ is not monotonic, while $e_3(N)$ obviously is. | |
| Jan 3, 2013 at 23:28 | history | edited | Yuichiro Fujiwara | CC BY-SA 3.0 |
added 558 characters in body; edited tags
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| Jan 2, 2013 at 19:42 | comment | added | user9072 | @Yuichiro Fujiwara: Great you found it interesting. If I find time to flesh it out a bit (and it is not superseeded) I certainly could repost it as answer later. [And, neither am I still mad at you nor was I ever. Actually, I just tried to be more friendly with this last sentence. You see, as I said, I am not so good at 'being friendly' or at least at being perceived as friendly.] | |
| Jan 2, 2013 at 19:22 | comment | added | Yuichiro Fujiwara | @Robert Thank you for the interesting results! Apparently, $e_3({\boldsymbol{Z}_n})$ is more irregular and thus has to be difficult to give a really sharp and general bound. Intriguing. | |
| Jan 2, 2013 at 19:18 | comment | added | Yuichiro Fujiwara | Wow. Thank you for the very informative comments, quid! You should've posted it as an answer. The general asymptotic case is as exciting as it is difficult, but I'm also interested in non-asymptotic results. So results on some restricted $n$ and computational math things are great, too. About the last sentence of your comment, you're still mad at me, aren't ya? | |
| Jan 2, 2013 at 19:05 | comment | added | Robert Israel | The results I get for $e_3(Z_n)$, $n=1,2,\ldots,30$ are $1, 2, 2, 2, 2, 4, 3, 4, 4, 4, 4, 4, 4, 6, 4, 6, 5, 8, 6, 8, 6, 8, 6, 8, 7, 8, 8, 8, 8, 8$. This sequence does not seem to be in the OEIS yet, though $e_3(N)$ is (sequence A003002). | |
| Jan 2, 2013 at 17:21 | comment | added | user9072 | This for the asymptotic results. If you care for explicit small values, things could/should be a bit different. Although I do not know anything right now. Sorry, for the strain of comments; but I did not want to give this as an answer and then I must not be too terse either :) | |
| Jan 2, 2013 at 17:18 | comment | added | user9072 | In general, the asymptotic bounds seem presently still far away from each other so that this effect on the constant typically I think does not receive much attention, and people use the inequality you mention. It is not clear to me how much improvement (if any) one could make (abstarctly) on this inequaliy or if this was tried/succeeded. But at least I think it is not used typically. And by abstractly I mean that there is no clear conjecture for the asymtotic size or structure of extremal sets; so one would have to work only with the defining properties. | |
| Jan 2, 2013 at 17:12 | comment | added | user9072 | Just some minor remarks you might be aware of already: the upper bound for the integer case is in fact a bound for the cyclic case plus (essentially) the inequality you mention. So, Sanders's paper is (also) the best known for the cyclic case (see the final section to notice this). You could use his tech. results directly for cyclic but all this would be absorbed in the constant of the big-Oh. I think this is still best (see Gowers's blog gowers.wordpress.com/2011/02/05/polymath6-a-is-to-b-as-c-is-to for discussion on project for improve. but it seems it didn't yet suceed). | |
| Jan 2, 2013 at 16:10 | history | asked | Yuichiro Fujiwara | CC BY-SA 3.0 |