  We present the characterization of metric spaces that are micro-, macro- or
bi-uniformly equivalent to the extended Cantor set
$\{\sum_{i=-n}^\infty\frac{2x_i}{3^i}:n\in\IN
,\;(x_i)_{i\in\IZ}\in\{0,1\}^\IZ\}\subset\IR$, which is bi-uniformly equivalent
to the Cantor bi-cube $2^{<\IZ}=\{(x_i)_{i\in\IZ}\in \{0,1\}^\IZ:\exists
n\;\forall i\ge n\;x_i=0\}$ endowed with the metric
$d((x_i),(y_i))=\max_{i\in\IZ}2^i|x_i-y_i|$. Those characterizations imply that
any two (uncountable) proper isometrically homogeneous ultrametric spaces are
coarsely (and bi-uniformly) equivalent. This implies that any two countable
locally finite groups endowed with proper left-invariant metrics are coarsely
equivalent. For the proof of these results we develop a technique of towers
which can have an independent interest.
