Directions sets: A generalisation of ratio sets

For every integer and every, we define the -directions sets of as and, where is the Euclidean norm and. Via an appropriate homeomorphism, is a generalisation of the ratio set. We study and as subspaces of. In particular, generalising a result of Bukor and Tóth, we provide a characterisation of the sets such that there exists satisfying, where denotes the set of accumulation points of. Moreover, we provide a simple sufficient condition for to be dense in. We conclude with questions for further research.