Finite phase transitions in countable abelian groups

Let A be an infinite set that generates a group G. The sphere SA(r) is the set of elements of G for which the word length with respect to A is exactly r. We say G admits all finite transitions if for every r ≥ 2 and every finite symmetric subset \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${W \subset G{\setminus}\{e\}}$$\end{document}, there exists an A with SA(r) = W. In this paper we determine which countable abelian groups admit all finite transitions. We also show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^n}$$\end{document} and the finitary symmetric group on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{N}}$$\end{document} admit all finite transitions.