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.

Directions sets: A generalisation of ratio sets / Leonetti, Paolo; Sanna, Carlo. - In: BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY. - ISSN 0004-9727. - STAMPA. - 101:3(2020), pp. 389-395. [10.1017/S0004972719000959]

Directions sets: A generalisation of ratio sets

Sanna Carlo
2020

Abstract

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2789380