Boundary entropy spectra as finite subsums

In this paper we provide a concrete construction of Furstenberg entropy values of $\tau$-boundaries of the group $\mathbb{Z}[\frac{1}{p_1},\ldots,\frac{1}{p_{l}}]\rtimes \{p_1^{n_1}\cdots p_{l}^{n_{l}} \, : \, n_i\in\mathbb{Z}\}$ by choosing an appropriate random walk $\tau$. We show that the boundary entropy spectrum can be realized as the subsum-set for any given finite sequence of positive numbers.