On Accumulation Points of Ratio Sets of Positive Integers

Bukor, József; Tóth, János T.

doi:10.1080/00029890.1996.12004776

Cited by 18 publications

(18 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The major open problem is to characterise all subsets of N whose ratio sets are dense in the positive real numbers. In this direction, many results have already been obtained, see for example [2,3,4,5,10,14,15,16,17,18,23,24,27,28]. An analogous study has also been done for algebraic number fields, see for example [7,9,26].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 91%

$p$-Adic quotient sets: diagonal forms

Antony¹,

Barman²,

Miska³

2022

Preprint

View full text Add to dashboard Cite

It is an open problem to study the denseness of R(A) in the p-adic numbers when A is the set of nonzero values attained by an integral form. This problem has been answered for quadratic forms. Very recently, Antony and Barman have studied this problem for the diagonal binary cubic forms ax 3 + by 3 , where a and b are integers. In this article, we study this problem for diagonal forms. We extend the results of Antony and Barman to the diagonal binary forms ax n + by n for all n ≥ 3. We also study p-adic denseness of quotients of nonzero values attained by diagonal forms of degree n ≥ 3, where gcd(n, p(p − 1)) = 1.

show abstract

Section: Introduction and Statement Of Resultsmentioning

confidence: 91%

$p$-Adic quotient sets: diagonal forms

Antony¹,

Barman²,

Miska³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Observe that in Example 1 we have α i /β i = δ 2 /(γ 2 a) and the minimum of the right hand side of (16) is the same as in (14) and (1). In Example 2, for x = α and y = β , the right hand side of (16) equals 0.229. .…”

Section: Extension Of Theoremmentioning

confidence: 87%

Asymptotic density of A⊂ ℕ and density of the ratio set R(A)

Strauch¹,

Tóth²

1998

Acta Arith.

View full text Add to dashboard Cite

“…The question of when R(A) is dense in R + is a classical topic and has been studied by many researchers (see, e.g., [1,2,3,7,8,9,11,15]).…”

Section: Introductionmentioning

confidence: 99%

“…Remark 1.15. In [5,Theorem 4.2] it is stated that R(S 3 2 ) is not dense in Q 3 . This is not correct, since R(S 3 2 ) is dense in Q 3 in light of Corollary 1.12.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the p-adic denseness of the quotient set of a polynomial image

Miska,

Murru,

Sanna

2018

Preprint

View full text Add to dashboard Cite

We consider the case in which A is the image of Z + under a polynomial f ∈ Z[X], and we give some conditions under which R(A) is dense in Qp. Then, we apply these results to determine when R(S n m ) is dense in Qp, where S n m is the set of numbers of the formwith x1, . . . , xm ≥ 0 integers. This allows us to answer a question posed in [Garcia et al., p-adic quotient sets, Acta Arith. 179, 163-184]. We end leaving an open question.

show abstract

On Accumulation Points of Ratio Sets of Positive Integers

Cited by 18 publications

References 4 publications

$p$-Adic quotient sets: diagonal forms

$p$-Adic quotient sets: diagonal forms

Asymptotic density of A⊂ ℕ and density of the ratio set R(A)

On the p-adic denseness of the quotient set of a polynomial image

Contact Info

Product

Resources

About