Light Subsets of N with Dense Quotient Sets

Hedman, Shawn; Rose, David A.

doi:10.4169/193009709x458618

Cited by 10 publications

(11 citation statements)

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“…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

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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.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 91%

$p$-Adic quotient sets: diagonal forms

Antony¹,

Barman²,

Miska³

2022

Preprint

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“…On the other hand, we have ) is dense in Q p . (e) On the one hand, by (5), the 2-adic valuation of each nonzero element of T 8 15 is congruent to 0, 1, 2, or 3 modulo 8. Hence, R(T…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…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%

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

Miska,

Murru,

Sanna

2018

Preprint

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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.

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“…The major open problem is to characterize all the subsets of Z (respectively, N) which are fractionally dense in R (respectively, R >0 ). In this direction, many results have already been obtained in [1], [2], [3], [4], [5] [9], [10], [11], [14], [15] [19], [20], [21] and [22]. This problem has also been considered in the p-adic set-up in [7], [8], [13] and [17].…”

Section: Introductionmentioning

confidence: 99%