Transcendency of some constants related to integer sequences of polynomial iterations

Dubickas, Artūras

doi:10.1007/s11139-021-00428-5

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…As a consequence of Theorem 1.4, we have the following result, which strengthens a result of Dubickas [4,Theorem 2] under the assumption that a…”

Section: Introductionsupporting

confidence: 84%

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Growth of linear combination of exponential sequences (over $\bar{\mathbb{Q}}$) Modulo $1$ and their consequences

Kumar¹

2022

Preprint

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Let k ∈ N, and let (α1, . . . , α k ) be a non-degenerate tuple of algebraic numbers with |αi| ≥ 1 for 1 ≤ i ≤ k and none of αi is root of unity. Let K be a number field and β ∈ (0, 1) be an algebraic number. Let θ ∈ (0, 1) be any number. In this article, we study the solutions of the inequalitydepending on β rational or irrational. Here ||x|| denotes the distance between x and its nearest integer in Z. As a consequence of our result, we show that ifis a polynomial of degree d ≥ 2 and (xn) is a sequence of integers satisfying xn+1 = P (xn) for all n = 0, 1, 2 . . . , and xn → ∞ as n → ∞, then either the number α := limn→∞ x d −n n is transcendental or a 1/d−1 d α d n is not a pseudo-Piost number for all large values of n, provided a d−1 da d is not an integer.

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“…As a consequence of Theorem 1.4, we have the following result, which strengthens a result of Dubickas [4,Theorem 2] under the assumption that a…”

Section: Introductionsupporting

confidence: 84%

“…These sequences can be found in the On-Line Encyclopedia of Integer Sequences [11](see also [1]). In [4], Dubickas proved the transcendence of these constants.…”

Section: Introductionmentioning

confidence: 99%

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Growth of linear combination of exponential sequences (over $\bar{\mathbb{Q}}$) Modulo $1$ and their consequences

Kumar¹

2022

Preprint

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Trees with minimum number of infima closed sets

Andriantiana

Wagner

2022

Discrete Mathematics

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The transcendence of growth constants associated with polynomial recursions

Kumar

2024

Int. J. Number Theory

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Let [Formula: see text], [Formula: see text], be a polynomial of degree [Formula: see text]. Let [Formula: see text] be a sequence of integers satisfying [Formula: see text] Set [Formula: see text]. Then, under the assumption [Formula: see text], in a recent result by [A. Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, Ramanujan J. 57 (2022) 569–581], either [Formula: see text] is transcendental or [Formula: see text] can be an integer or a quadratic Pisot unit with [Formula: see text] being its conjugate over [Formula: see text]. In this paper, we study the nature of such [Formula: see text] without the assumption that [Formula: see text] is in [Formula: see text], and we prove that either the number [Formula: see text] is transcendental, or [Formula: see text] is a Pisot number with [Formula: see text] being the order of the torsion subgroup of the Galois closure of the number field [Formula: see text]. Other results presented in this paper investigate the solutions of the inequality [Formula: see text] in [Formula: see text], considering whether [Formula: see text] is rational or irrational. Here, [Formula: see text] represents a number field, and [Formula: see text]. The notation [Formula: see text] denotes the distance between [Formula: see text] and its nearest integer in [Formula: see text].

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Transcendency of some constants related to integer sequences of polynomial iterations

Cited by 3 publications

References 9 publications

Growth of linear combination of exponential sequences (over $\bar{\mathbb{Q}}$) Modulo $1$ and their consequences

Growth of linear combination of exponential sequences (over $\bar{\mathbb{Q}}$) Modulo $1$ and their consequences

Trees with minimum number of infima closed sets

The transcendence of growth constants associated with polynomial recursions

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