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EnglishThe ratio set of a set of positive integers A is defined as R(A):={a/b:a,b∈A}. The study of the denseness of R(A) in the set of positive real numbers is a classical topic, and, more recently, the denseness in the set of p-adic numbers Qp has also been investigated. Let A1,…,Ak be a partition of N into k sets. We prove that for all prime numbers p but at most ⌊log2k⌋ exceptions at least one of R(A1),…,R(Ak) is dense in Qp. Moreover, we show that for all prime numbers p but at most k−1 exceptions at least one of A1,…,Ak is dense in Zp. Both these results are optimal in the sense that there exist partitions A1,…,Ak having exactly ⌊log2k⌋, respectively, k−1, exceptional prime numbers; and we give explicit constructions for them. Furthermore, as a corollary, we answer negatively a question raised by Garcia et al.
p-adic denseness of members of partitions of N and their ratio sets / Miska, Piotr; Sanna, Carlo. - In: BULLETIN OF THE MALAYSIAN MATHEMATICAL SOCIETY. - ISSN 0126-6705. - STAMPA. - 43(2020), pp. 1127-1133. [10.1007/s40840-019-00728-6]
| Titolo: | p-adic denseness of members of partitions of N and their ratio sets | |
| Autori: | ||
| Data di pubblicazione: | 2020 | |
| Rivista: | ||
| Digital Object Identifier (DOI): | http://dx.doi.org/10.1007/s40840-019-00728-6 | |
| Appare nelle tipologie: | 1.1 Articolo in rivista |
File in questo prodotto:
| File | Descrizione | Tipologia | Licenza | |
|---|---|---|---|---|
| denspart.pdf | 2. Post-print / Author's Accepted Manuscript | PUBBLICO - Tutti i diritti riservati | Visibile a tuttiVisualizza/Apri | |
| p-Adic Denseness of Members of Partitions of NN.pdf | 2a Post-print versione editoriale / Version of Record | Non Pubblico - Accesso privato/ristretto | Administrator Richiedi una copia |
http://hdl.handle.net/11583/2818867
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