Timeline for Arithmetic progressions in Van der Waerden's theorem
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Apr 27, 2017 at 7:17 | comment | added | Xiaoyu He | Not too much to add to that answer - essentially what it shows is such a set $S_x$ will have long arithmetic progressions of common difference $d$ iff $xd$ is very close to an integer, and the length will be inversely proportional to how close this is. In particular the example $x=\phi$ should be best possible up to constants with $\{xd\} \ge d^{-1}$ by standard Diophantine approximation results, and so $d_{\ell}$ growing linearly in $\ell$ is best possible using this construction. | |
| Apr 27, 2017 at 3:38 | comment | added | Andy | Thanks! The link you gave suggests the following idea which is quite intuitive. If $x\in [0,1]$ is irrational, let $S = \{n\in \mathbb{Z}: xn-\lfloor xn \rfloor\in [0, 1/2]\}$. Then this clearly has the property I want. How fast can one make the function $\ell\to d_\ell$ grow? | |
| Apr 27, 2017 at 3:35 | vote | accept | Andy | ||
| Apr 27, 2017 at 3:34 | vote | accept | Andy | ||
| Apr 27, 2017 at 3:35 | |||||
| Apr 27, 2017 at 2:13 | history | undeleted | Xiaoyu He | ||
| Apr 27, 2017 at 2:12 | history | edited | Xiaoyu He | CC BY-SA 3.0 |
deleted 378 characters in body
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| Apr 27, 2017 at 1:31 | history | deleted | Xiaoyu He | via Vote | |
| Apr 27, 2017 at 1:31 | history | answered | Xiaoyu He | CC BY-SA 3.0 |