## Rocky Mountain Journal of Mathematics

### On fractionally dense sets

#### Abstract

In this article, we prove that some subsets of the set of natural numbers $\mathbb {N}$ and any non-zero ideals of an order of an imaginary quadratic field are fractionally dense in $\mathbb {R}_{>0}$ and $\mathbb {C}$, respectively.

#### Article information

Source
Rocky Mountain J. Math., Volume 49, Number 3 (2019), 743-760.

Dates
First available in Project Euclid: 23 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1563847231

Digital Object Identifier
doi:10.1216/RMJ-2019-49-3-743

Mathematical Reviews number (MathSciNet)
MR3983298

Zentralblatt MATH identifier
07088334

Subjects
Primary: 11A41: Primes 11B05: Density, gaps, topology

#### Citation

Chattopadhyay, Jaitra; Roy, Bidisha; Sarkar, Subha. On fractionally dense sets. Rocky Mountain J. Math. 49 (2019), no. 3, 743--760. doi:10.1216/RMJ-2019-49-3-743. https://projecteuclid.org/euclid.rmjm/1563847231

#### References

• B. Brown, M. Dairyko, S.R. Garcia, B. Lutz and M. Someck, Four quotient set gems, Amer. Math. Month. 121 (2014), 590–599.
• J. Bukor, P. Erdős, T. Šalát and J.T. Tóth, Remarks on the $R$-density of sets of numbers, II, Math. Slov. 47 (1997), 517–256.
• J. Bukor, T. Šalát and J.T. Tóth, Remarks on $R$-density of sets of numbers, Tatra Mtn. Math. Publ. 11 (1997), 159–165.
• J. Bukor and J.T. Tóth, On accumulation points of ratio sets of positive integers, Amer. Math. Month. 103 (1996), 502–504.
• ––––, On some criteria for the density of the ratio sets of positive integers, JP J. Alg. Num. Th. Appl. 3 (2003), 277–287.
• S.R. Garcia, Quotients of Gaussian primes, Amer. Math. Month. 120 (2013), 851–853.
• S.R. Garcia, Y.X. Hong, F. Luca, E. Pinsker, C. Sanna, E. Schechter and A. Starr, $p$-adic quotient sets, Acta Arith. 179 (2017), 163–184.
• S.R. Garcia and F. Luca, Quotients of Fibonacci numbers, Amer. Math. Month. 123 (2016), 1039–1044.
• S.R. Garcia, V. Selhorst-Jones, D.E. Poore and N. Simon, Quotient sets and Diophantine equations, Amer. Math. Month. 118 (2011), 704–711.
• S. Hedman and D. Rose, Light subsets of $\mathbb{N}$ with dense quotient sets, Amer. Math. Month. 116 (2009), 635–641.
• D. Hobby and D.M. Silberger, Quotients of primes, Amer. Math. Month. 100 (1993), 50–52.
• T.A. Hulse and M. Ram Murty, Bertrand's postulates for number fields, Colloq. Math. 147 (2017), 165–180.
• P. Miska and C. Sanna, $p$-adic denseness of members of partitions of $\mathbb{N}$ and their ratio sets, doi.10.13140/RG.2.2.28136.57608.
• T. Šalát, On ratio sets of natural numbers, Acta Arith. 15 (1969), 273–278.
• T. Šalát, Corrigendum to the paper, On ratio sets of sets of natural numbers, Acta Arith. 16 (1969/1970), 103.
• ––––, Quotientbasen und $(R)$-dichte Mengen, Acta Arith. 19 (1971), 63–78.
• C. Sanna, The quotient set of $k$-generalised Fibonacci numbers is dense in $\mathbb{Q}_{p}$, Bull. Australian Math. Soc. 96 (2017), 24–29.
• B.D. Sittinger, Quotients of primes in a quadratic number ring, Not. Num. Th. Discr. Math. 24 (2018), 55–62.
• P. Starni, Answers to two questions concerning quotients of primes, Amer. Math. Month. 102 (1995), 347–349.
• O. Strauch, Distribution functions of ratio sequences, An expository paper, Tatra Mtn. Math. Publ. 64 (2015), 133–185.
• O. Strauch and J.T. Tóth, Asymptotic density of $A \subset \mathbb{N}$ and density of the ratio set, Acta Arith. 87 (1998), 67–78.
• ––––, Corrigendum to Theorem $5$ of the paper, Asymptotic density of $A \subset \mathbb{N}$ and density of the ratio set $R(A)$, Acta Arith. 103 (2002), 191–200.