Timeline for Simultaneous small fractional parts of polynomials
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 15, 2020 at 18:05 | vote | accept | Joe Previdi | ||
| Jul 15, 2020 at 17:45 | answer | added | Joe Previdi | timeline score: 2 | |
| Jul 15, 2020 at 17:00 | comment | added | Joe Previdi | @NoamD.Elkies Would you be able to point to a resource proving that $\mathbb{Q}$-linearly dependent collections are equidistributed on a closed subgroup? | |
| Jul 15, 2020 at 8:30 | history | edited | Joe Previdi | CC BY-SA 4.0 |
Updates based on the comment section; progress on how equidistribution only works for linearly independent collections
|
| Jul 12, 2020 at 13:45 | comment | added | Noam D. Elkies | @Fedor Petrov: There's a natural generalization from the circle ${\bf R} / {\bf Z}$ to the torus ${\bf R}^m / {\bf Z}^m$. A sequence of vectors $v_n \in {\bf R}^m / {\bf Z}^m$ is asymptotically equidistributed iff for even nonzero $a \in {\bf Z}^m$ we have $$ \sum_{n=1}^N \exp 2\pi i (a \cdot v_n) = o(N) $$ as $N \to \infty$. This does in the end come to the estimate for a single polynomial. (If $p_1,\ldots,p_m$ are $\bf Q$-linearly dependent then they're equidistributed in some closed subgroup of the torus.) | |
| Jul 12, 2020 at 10:26 | comment | added | Fedor Petrov | So it is equivalent to the analogous question about several polynomials of the form $cx^k$, $k>0$, right? | |
| Jul 12, 2020 at 10:22 | comment | added | Fedor Petrov | @NoamD.Elkies do you mean Weyl's criterion for a single polynomial? | |
| Jul 12, 2020 at 2:47 | comment | added | Noam D. Elkies | OK, but that must be implied by Weyl's equidistribution criterion on a torus. Schmidt surely knows this, so I expect that there's some other issue, such as speed of convergence. | |
| Jul 12, 2020 at 2:36 | comment | added | Joe Previdi | *distance to the nearest integer | |
| Jul 12, 2020 at 2:36 | history | edited | Joe Previdi | CC BY-SA 4.0 |
wrong function, should be distance to nearest integer
|
| Jul 12, 2020 at 2:26 | history | edited | Joe Previdi | CC BY-SA 4.0 |
edited body
|
| Jul 12, 2020 at 2:15 | history | edited | Joe Previdi | CC BY-SA 4.0 |
replaced fractional part function with nearest integer
|
| Jul 12, 2020 at 2:13 | comment | added | Joe Previdi | Great catch. This shouldn't be fractional part, but the nearest integer function instead. | |
| Jul 12, 2020 at 1:58 | comment | added | Noam D. Elkies | OK, but something still seems to be missing. For example, if $p_1$ never takes integer values on $\mathbb N$, and $p_2 = -p_1$, then $\{ p_1(n) \} + \{ p_2(n) \} = 1$ for all $n \in \mathbb N$, so the fractional parts cannot both be small. | |
| Jul 12, 2020 at 0:09 | history | edited | Joe Previdi | CC BY-SA 4.0 |
added proper restriction on polynomials
|
| Jul 12, 2020 at 0:09 | comment | added | Joe Previdi | Ah of course, sorry! It is imposed that the polynomials have integer constant term. I will edit the post. | |
| Jul 12, 2020 at 0:06 | comment | added | Noam D. Elkies | Not if each polynomial is of the form $P(x) + 1/2$ for some $P \in {\bf Z}[X]$. So some hypothesis is necessary. What is condition imposed in the $p_i$ in your source? | |
| Jul 11, 2020 at 23:51 | history | asked | Joe Previdi | CC BY-SA 4.0 |