Coarse structures on groups defined by $T$-sequences

A sequence $(a_{n})$ in an Abelian group is called a $T$-sequence if there exists a Hausdorff group topology on $G$ in which $(a_{n})$ converges to $0$. For a $T$-sequence $(a_{n})$, $\tau_{(a_{n}) }$ denotes the strongest group topology on $G$ in which $(a_{n})$ converges to $0$. The ideal $\mathcal{I}_{(a_{n})}$ of all precompact subsets of $(G, \tau_{(a_{n}) } )$ defines a coarse structure on $G$ with base of entourages $\{(x, y): x-y \in P \}$, $P\in\mathcal{I}_{(a_{n})}.$ We prove that $asdim \ \ (G, \mathcal{I}_{(a_{n}) }) =\infty$ for every non-trivial $T$-sequence $(a_{n})$ on $G$, and the coarse group $(G, \mathcal{I}_{(a_{n}) })$ has 1 end provided that $(a_{n})$ generates $G$. The keypart play asymorphic copies of the Hamming space in $(G, \mathcal{I}_{(a_{n})})$.