Timeline for If I exchange infinitely many digits of $\pi$ and $e$, are the two resulting numbers transcendental?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 25, 2017 at 18:07 | comment | added | Gerhard Paseman | Could you spell that out, @dhy? If you put it in an answer, we might be able to use it even if it doesn't' work out that way. (And I think the measure 0 result means a likely yes answer, not a no answer, to Erin's question.) Gerhard "Hunting Old And New Ideas" Paseman, 2017.03.25. | |
| Mar 25, 2017 at 17:52 | comment | added | dhy | I believe a slightly more complicated argument will in fact show that the set of $(a,b)$ in $\mathbb{R}x\mathbb{R}$ from which an algebraic number can be formed by swapping digits is of measure $0$, which means the answer to the original problem is likely no. | |
| Mar 24, 2017 at 2:21 | comment | added | user10290 | I did imagine a universe where we know about a bunch of transcendentals in that we know they are coded by $e$ and $\pi$. I thought maybe we could force it to be true and find out it was already true in the ground model. But just having a forcing extension, now that I see it is difficult, would be great. | |
| Mar 24, 2017 at 2:17 | comment | added | user10290 | Thanks for more ways to generate transcendental numbers! | |
| Mar 24, 2017 at 2:16 | comment | added | Gerhard Paseman | Indeed, I think you can, by organizing the algebraics by place value and then constructing the list so that you end up not skipping any. This might answer her question positively. Gerhard "Or Present A New Paradox" Paseman, 2017.03.23. | |
| Mar 24, 2017 at 2:11 | comment | added | Gerhard Paseman | Can you reverse this argument? Given $\pi$ and $e$, can you find an enumeration of algebraics that guarantees the rearrangements will be transcendental? And if not in this universe, then in some other? Gerhard "Maybe Erin Meant That Question" Paseman, 2017.03.23. | |
| Mar 24, 2017 at 1:36 | comment | added | Gerhard Paseman | You can probably extend this to more numbers with care. (Unless you are Joel, in which case you can also do it with abandon.) Gerhard "Reduce And Reuse And Recycle" Paseman, 2017.03.23. | |
| Mar 24, 2017 at 1:35 | comment | added | Joel David Hamkins | A similar argument produces an infinite list of such numbers. Indeed, one can have continuum many such numbers: consider the numbers whose even digits diagonalize (uniformly) against the algbraic numbers, and whose odd digits are arbitrary. Swapping digits on on two of these on an arbitrary set will still diagonalize against the algebraics, and hence will be transcendental. | |
| Mar 24, 2017 at 1:22 | history | answered | Gerhard Paseman | CC BY-SA 3.0 |