Timeline for Irrationality measure of powers
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Nov 14, 2021 at 5:37 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
a minor typo
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| Nov 11, 2021 at 22:01 | vote | accept | joaopa | ||
| Nov 9, 2021 at 7:40 | comment | added | Dave L Renfro | FYI, the same dichotomy of Leb. measure and Baire category for irrationality measure holds for normal numbers -- Lebesgue-almost-all real numbers have irrationality measure $2$ (min. value, when the inconsequential countably many alg. numbers are omitted) and are normal (minimum digit frequency variation), and Baire-almost-all real numbers have irrationality measure $\infty$ and are highly non-normal (see Olsen and Stylianou), so this answer might be of interest. | |
| Nov 9, 2021 at 7:27 | comment | added | Dave L Renfro | I wonder why the downvote. Maybe I'm missing something, but I don't see anything here that would merit downvoting, even if this states what some might consider as relatively obvious and well-known -- both of which are highly dependent on the reader and thus within the scope of one of the original purposes of this site. Regarding your last sentence, it's also worth recalling that almost all (in the sense of Baire category) irrational numbers (or even real numbers) have irrationality measure $\infty,$ so their abundance depends on the notion of largeness one uses. | |
| Oct 14, 2020 at 19:54 | comment | added | Manuel Norman | That's an interesting question. I think that it might be possible. Maybe an upper bound can be derived using the limit formula involving the convergents of the simple continued fractions (of the powers, in this case) found by Sondow. This is just an idea - I have not checked if it works | |
| Oct 14, 2020 at 19:03 | comment | added | joaopa | Thank you for the case $\alpha$ irationnal. In the transcendental case, at least can one obtain a upper bound of $\mu(\alpha^n)$ depending on $\mu(\alpha)$? | |
| Oct 14, 2020 at 18:13 | history | answered | Manuel Norman | CC BY-SA 4.0 |