Do you know of any volume estimates for the Haar measure on a compact connected metrizable abelian group $G$?
Specifically, is it true that there is $s>0$ and $C>1$ so that, for all $g\in G$ and $0<r<diam(G)$, we have $$C^{-1}r^s\leq vol_G(B_d(g,r))\leq Cr^s,$$ where $B_d(g,r)$ denotes the closed ball w.r.t a translation-invariant metric $d$ of G?
I can see that this holds for the $n$-torus (with $s=n$) since the Haar measure is the $n$-dimensional Lebesgue measure pushed-forwarded via the quotient map $\mathbb R^n\to \mathbb R^n/\mathbb Z^n$. However, I cannot tell in general.
Many thanks!