DIRECTIONS SETS: A GENERALISATION OF RATIO SETS

Abstract

For every integer $k\geq 2$ and every $A\subseteq \mathbb{N}$, we define the $k$-directions sets of $A$ as $D^{k}(A):=\{\boldsymbol{a}/\Vert \boldsymbol{a}\Vert :\boldsymbol{a}\in A^{k}\}$ and $D^{\text{}\underline{k}}(A):=\{\boldsymbol{a}/\Vert \boldsymbol{a}\Vert :\boldsymbol{a}\in A^{\text{}\underline{k}}\}$, where $\Vert \cdot \Vert$ is the Euclidean norm and $A^{\text{}\underline{k}}:=\{\boldsymbol{a}\in A^{k}:a_{i}\neq a_{j}\text{ for all }i\neq j\}$. Via an appropriate homeomorphism, $D^{k}(A)$ is a generalisation of the ratio set$R(A):=\{a/b:a,b\in A\}$. We study $D^{k}(A)$ and $D^{\text{}\underline{k}}(A)$ as subspaces of $S^{k-1}:=\{\boldsymbol{x}\in [0,1]^{k}:\Vert \boldsymbol{x}\Vert =1\}$. In particular, generalising a result of Bukor and Tóth, we provide a characterisation of the sets $X\subseteq S^{k-1}$ such that there exists $A\subseteq \mathbb{N}$ satisfying $D^{\text{}\underline{k}}(A)^{\prime }=X$, 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 with questions for further research.