Directions sets: A generalization of ratio sets

For every integer $k\ge 2$ and every $A\subseteq \mathbb{N}$, we define the \emph{$k$-directions sets} of $A$ as $D^k(A) := {{\bf a} / |{\bf a}| : {\bf a} \in A^k}$and$D^{\underline{k}}(A) := {{\bf a} / |{\bf a}| : {\bf a} \in A^{\underline{k}}}$,where$|\cdot|$istheEuclideannormand$A^{\underline{k}} := {{\bf a} \in A^k : a_i \neq a_j $\text{ for all }$ i \neq j}$\mathrm{.}Viaanappropriatehomeomorphism,$D^k(A)$ is a generalization of the \emph{ratio set} $R(A):=\{a\mathrm{/}b:a,b\in A\}$, which has been studied by many authors. We study $D^k(A)$ and $D^{\underset{‾}{k}}(A)$ as subspaces of $S^{k-1}:=\{\mathbf{x}\in [0,1{]}^k:\mathrm{\parallel }\mathbf{x}\mathrm{\parallel }=1\}$. In~particular, generalizing a result of Bukor and T'oth, we provide a characterization of the sets $X\subseteq S^{k-1}$ such that there exists $A\subseteq \mathbb{N}$ satisfying $D^{\underset{‾}{k}}(A{)}^{\mathrm{\prime }}=X$, where $Y^{\mathrm{\prime }}$ denotes the set of accumulation points of $Y$. Moreover, we provide a simple sufficient condition for $D^k(A)$ to be dense in $S^{k-1}$. We conclude leaving some questions for further research.