$p$-Adic quotient sets: diagonal forms

Antony, Deepa; Barman, Rupam; Miska, Piotr

doi:10.48550/arxiv.2201.04372

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2022

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“…Very recently, the denseness of ratio sets in the p-adic completion Q p have also been considered (cf. [1], [2], [12], [13], [18], [22]).…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

On denseness of certain direction and generalized direction sets

Antony¹,

Barman²,

Chattopadhyay³

2022

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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 k-generalized direction sets and distinct k-generalized direction sets for subsets of positive integers. We prove a necessary condition for a subset of S k−1 := {x ∈ [0, 1] k : ||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 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.

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“…Very recently, the denseness of ratio sets in the p-adic completion Q p have also been considered (cf. [1], [2], [12], [13], [18], [22]).…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%