Directions sets: A generalization of ratio sets

Abstract

For every integer $k\ge 2$ and every $A\subseteq \mathbb{N}$, we define the \emph{$k$-directions sets} of $A$ as $D^k(A):=\{\mathbf{a}/\Vert \mathbf{a}\Vert :\mathbf{a}\in A^k\}$ and $D^{\underline{k}}(A):=\{\mathbf{a}/\Vert \mathbf{a}\Vert :\mathbf{a}\in A^{\underline{k}}\}$, where $\Vert \cdot \Vert$ is the Euclidean norm and $A^{\underline{k}}:=\{\mathbf{a}\in A^k:a_i\ne a_j\text{ for all }i\ne j\}$. Via an appropriate homeomorphism, $D^k(A)$ is a generalization of the \emph{ratio set} $R(A):=\{a/b:a,b\in A\}$, which has been studied by many authors. We study $D^k(A)$ and $D^{\underline{k}}(A)$ as subspaces of $S^{k-1}:=\{\mathbf{x}\in [0,1{]}^k:\Vert \mathbf{x}\Vert =1\}$. In~particular, generalizing a result of Bukor and Tóth, we provide a characterization of the sets $X\subseteq S^{k-1}$ such that there exists $A\subseteq \mathbb{N}$ satisfying $D^{\underline{k}}(A{)}^{\prime }=X$, where $Y^{\prime }$ 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.

7 pp. Accepted in Bull. Austr. Math. Soc