On spectral synthesis on zero-dimensional Abelian groups

Let $G$ be a zero-dimensional locally compact Abelian group all of whose elements are compact, and let $C(G)$ be the space of all complex-valued continuous functions on $G$. A closed linear subspace $\mathscr H\subseteq C(G)$ is said to be an invariant subspace if it is invariant with respect to the translations $\tau_y\colon f(x)\mapsto f(x+y)$, $y\in G$. In the paper, it is proved that any invariant subspace $\mathscr H$ admits spectral synthesis, that is, $\mathscr H$ coincides with the closed linear span of the characters of $G$ belonging to $\mathscr H$.
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