Connected neighborhoods in Cartesian products of solenoids

Given a collection of pairwise co-prime integers $$, greater than $1$, we consider the product $\Sigma =\Sigma {m{1}}\times \cdots \times \Sigma {m{r}}$,whereeach$\Sigma {m{i}}$isthe$m_{i}$-adic solenoid. Answering a question of D. P. Bellamy and J. M. \L ysko, in this paper we prove that if $M$ is a subcontinuum of $\mathrm{\Sigma }$ such that the projections of $M$ on each ${\mathrm{\Sigma }}_{m_i}$ are onto, then for each open subset $U$ in $\mathrm{\Sigma }$ with $M\subset U$, there exists an open connected subset $V$ of $\mathrm{\Sigma }$ such that $M\subset V\subset U$; i.e. any such $M$ is ample in the sense of Prajs and Whittington [10]. This contrasts with the property of Cartesian squares of fixed solenoids $\Sigma {m{i}}\times \Sigma {m{i}}$, whose diagonals are never ample [1].