On denseness of certain direction and generalized direction sets

TL;DR

This paper introduces generalized direction sets for positive integers, establishes conditions for their denseness in spheres, and explores their properties, partially answering a question by Leonetti and Sanna and extending to algebraic number fields.

Contribution

It generalizes the concept of direction sets, provides criteria for their denseness, and extends the analysis to algebraic number fields, advancing understanding of their structure.

Findings

Necessary condition for accumulation points of generalized direction sets.

Sufficient conditions for denseness in specific cases.

Partial resolution of a question by Leonetti and Sanna.

Abstract

Direction sets, recently introduced by Leonetti and Sanna, are generalization of ratio sets of subsets of positive integers. In this article, we generalize the notion of direction sets and define {\it $k$-generalized direction sets} and {\it distinct $k$-generalized direction sets} for subsets of positive integers. We prove a necessary condition for a subset of $\mathcal{S}^{k - 1} := \{\underline{x} \in [0,1]^{k} : ||\underline{x}|| = 1\}$ to be realized as the set of accumulation points of a distinct $k$-generalized direction set. We provide sufficient conditions for some particular subsets of positive integers so that the corresponding $k$-generalized direction sets are dense in $\mathcal{S}^{k - 1}$. We also consider the denseness properties of certain direction sets and give a partial answer to a question posed by Leonetti and Sanna. Finally we consider a similar question in the framework of an algebraic number field.