MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Consider an ultrametic space $X=K^n$ (for the norm $\| x\| =\max_{i=1,n} (\mid {x_1}\mid,\dots,\mid{x_n}\mid)$ where $K$ is an ultrametric field. Let $B(1):=\lbrace x \in X \mid \|x\| \leq 1\rbrace$ be the unit ball. Equip $X$ with Haar measure. Is it possible to partition $B(1)$ into $k$ smaller balls $B_{a_1}(r_1),\dots, B_{a_k}(r_k)$ (where $B_a(r)$ is the closed ball of radius $r$ and center $a$), $k>1$?

Or put in a different way: is there a measure preserving bianalytic map taking $B(1)$ to the union of $B_{a_i}(r_i)$?

Thank you

share|cite|improve this question
up vote 1 down vote accepted

If you can do the case of $K$ itself, decompose into $k$ smaller balls, then for $K^n$ you can decompose into $k^n$ smaller balls. For some ultrametric fields this works. Indeed, you say "Haar measure" so you must be assuming local compactness, and it works for any such field, since they are finite extensions of $p$-adic fields.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.