Miska, Piotr A note on \(p\)-adic denseness of quotients of values of quadratic forms. (English) Zbl 1470.11056 Indag. Math., New Ser. 32, No. 3, 639-645 (2021). A classification of nonsingular quadratic forms \(Q\) with integral coefficients such that the set of quotients of values of \(Q\) attained for integer arguments is dense in the field of \(p\)-adic numbers has been obtained by C. Donnay et al. [J. Number Theory 201, 23–39 (2019; Zbl 1459.11025)]. The paper under review gives another proof of the classification. Reviewer: Meinhard Peters (Münster) Cited in 3 Documents MSC: 11E08 Quadratic forms over local rings and fields 11B05 Density, gaps, topology Keywords:denseness; \(p\)-adic topology; quadratic form; quotient set; ratio set Citations:Zbl 1459.11025 PDFBibTeX XMLCite \textit{P. Miska}, Indag. Math., New Ser. 32, No. 3, 639--645 (2021; Zbl 1470.11056) Full Text: DOI arXiv References: [1] Brown, B.; Dairyko, M.; Garcia, S. R.; Lutz, B.; Someck, M., Four quotient set gems, Amer. Math. Monthly, 121, 7, 590-599 (2014) · Zbl 1371.11016 [2] Bukor, J.; Csiba, P., On estimations of dispersion of ratio block sequences, Math. Slovaca, 59, 3, 283-290 (2009) · Zbl 1212.11017 [3] Bukor, J.; Erdős, P.; Šalát, T.; Tóth, J. T., Remarks on the (R)-density of sets of numbers, II, Math. Slovaca, 47, 5, 517-526 (1997) · Zbl 0939.11005 [4] Bukor, J.; Šalát, T.; Tóth, J. T., Remarks on R-density of sets of numbers, (Number Theory, Liptovsk Ján, 1995. Number Theory, Liptovsk Ján, 1995, Tatra Mt. Math. Publ., vol. 11 (1997)), 159-165 · Zbl 0978.11003 [5] Bukor, J.; Tóth, J. T., On accumulation points of ratio sets of positive integers, Amer. Math. Monthly, 103, 6, 502-504 (1996) · Zbl 0857.11004 [6] Donnay, Ch.; Garcia, S. R.; Rouse, J., \(p\)-adic quotient sets II: Quadratic forms, J. Number Theory, 201, 23-39 (2019) · Zbl 1459.11025 [7] Garcia, S. R.; Hong, Y. X.; Luca, F.; Pinsker, E.; Sanna, C.; Schechter, E.; Starr, A., \(p\)-adic quotient sets, Acta Arith., 179, 2, 163-184 (2017) · Zbl 1428.11023 [8] Garcia, S. R.; Luca, F., Quotients of Fibonacci numbers, Amer. Math. Monthly, 123, 10, 1039-1044 (2016) · Zbl 1391.11027 [9] Garcia, S. R.; Poore, D. E.; Selhorst-Jones, V.; Simon, N., Quotient sets and Diophantine equations, Amer. Math. Monthly, 118, 8, 704-711 (2011) · Zbl 1254.11012 [10] Hedman, Sh.; Rose, D., Light subsets of N with dense quotient sets, Amer. Math. Monthly, 116, 7, 635-641 (2009) · Zbl 1229.11019 [11] Hobby, D.; Silberger, D. M., Quotients of primes, Amer. Math. Monthly, 100, 1, 50-52 (1993) · Zbl 0777.11001 [12] Miska, P.; Murru, N.; Sanna, C., On the \(p\)-adic denseness of the quotient set of a polynomial image, J. Number Theory, 197, 218-227 (2019) · Zbl 1450.11005 [13] Šalát, T., On ratio sets of natural numbers, Acta Arith., 15, 273-278 (1968) · Zbl 0177.07001 [14] Šalát, T., Corrigendum to the paper “On ratio sets of natural numbers”, Acta Arith., 16, 103 (1969) [15] Sanna, C., The quotient set of \(k\)-generalized Fibonacci numbers is dense in \(\mathbb{Q}_p\), Bull. Aust. Math. Soc., 96, 1, 24-29 (2017) · Zbl 1425.11031 [16] Serre, J.-P., (A Course in Arithmetic. A Course in Arithmetic, Graduate Texts in Mathematics, vol. 7 (1973), Springer-Verlag: Springer-Verlag New York-Heidelberg), translated from the French · Zbl 0256.12001 [17] Starni, P., Answers to two questions concerning quotients of primes, Amer. Math. Monthly, 102, 4, 347-349 (1995) · Zbl 0828.11004 [18] Strauch, O.; Tóth, J. T., Asymptotic density of \(A \subset \mathbb{N}\) and density of the ratio set \(R ( A )\), Acta Arith., 87, 1, 67-78 (1998) · Zbl 0923.11027 [19] Strauch, O.; Tóth, J. T., Corrigendum to Theorem 5 of the paper: “Asymptotic density of \(A \subset \mathbb{N}\) and density of the ratio set \(R ( A )\)”, Acta Arith. 87 (1) (1998) 67-78, Acta Arith., 103, 2, 191-200 (2002) · Zbl 0923.11027 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.