Timeline for Sums of powers of measures of $p$-adic balls
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Jan 11, 2023 at 12:34 | comment | added | Daniel Loughran | Very nice example, thanks. $\mathbb{Z}_p$ has the funny property that a disjoint unions of balls can sometimes be written a single ball. It may be the case that refining my balls to obtain a "minimal" representation of each ball, the result is still true. Any idea about this? | |
| Jan 10, 2023 at 16:48 | comment | added | YCor | @ChrisWuthrich well, I guess not. I basically used that the convergent sum $\sum 1/n^2$ has diverging sum of square roots, adapting to the given setting. But probably starting with $\sum 1/n^s$ with $s>1$ can be used, with a little more work, to show that no $\varepsilon<1$ works. | |
| Jan 10, 2023 at 16:46 | comment | added | Chris Wuthrich | This shows that you cannot hope it to converge for all $\varepsilon >0$, e.g. $\varepsilon\leq 1/2$ won't work for $p=2$. I guess the question remains if there exists a $\varepsilon<1$ for which it converges. | |
| Jan 10, 2023 at 16:24 | history | answered | YCor | CC BY-SA 4.0 |