If $K$ is a compact neighborhood of $0$, then $K \supseteq S(a, 0) = a\mathbb{Z}$. Since $S(a, 0)$ is clopen, it is a closed subspace of $K$, hence compact.

But $S(a, 0) \cong (\mathbb{Z}, \text{Furstenberg})$, via

and $(\mathbb{Z}, \text{Furstenberg})$ is not compact: the cover

has no finite subcover (any finite subfamily misses $p_1 p_2 \cdots p_k$ for primes outside the chosen set). Contradiction.