Boundary entropy spectra as finite subsums

TL;DR

This paper constructs specific Furstenberg entropy values for certain group boundaries using tailored random walks, demonstrating that the boundary entropy spectrum can be realized as subsum-sets of finite positive sequences.

Contribution

It provides a concrete method to realize boundary entropy spectra as subsum-sets for a class of groups, linking entropy values to finite sequences.

Findings

Boundary entropy spectra can be realized as subsum-sets.

Constructs explicit random walks with prescribed entropy spectra.

Connects entropy spectra to finite positive sequences.

Abstract

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.