A note on the transcendence of infinite products

Hančl, Jaroslav; Kolouch, Ondřej; Pulcerová, Simona; Štěpnička, Jan

doi:10.1007/s10587-012-0053-2

Cited by 4 publications

(2 citation statements)

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“…It is important to note that Corvaja and Hančl [2], in 2007, proved the transcendence of infinite product using the new diophantine approximation result proved by Corvaja and Zannier [3] for the first time in the literature. As an example, with the similar spirits in [4], one can conclude the following transcendental result.…”

Section: Introductionsupporting

confidence: 66%

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On simultaneous approximation of algebraic numbers

Kumar

Thangadurai

2022

Mathematika

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Let Γ ⊂ ℚ × be a finitely generated multiplicative group of algebraic numbers. Let 𝛼 1 , … , 𝛼 𝑟 ∈ ℚ × be algebraic numbers which are ℚ-linearly independent and let 𝜖 > 0 be a given real number. One of the main results that we prove in this article is as follows: There exist only finitely many tuplesPisot number for some integer 𝑖 ∈ {1, … , 𝑟} and 0 < |𝛼 𝑗 𝑞𝑢 − 𝑝 𝑗 | < 1 𝐻 𝜖 (𝑢)|𝑞| 𝑑 𝑟 +𝜀for all integers 𝑗 = 1, 2, … , 𝑟, where 𝐻(𝑢) is the absolute Weil height. In particular, when 𝑟 = 1, this result was proved by Corvaja and Zannier in [Acta Math. 193 (2004), 175-191]. As an application of our result, we also prove a transcendence criterion which generalizes a result of Hančl, Kolouch, Pulcerová, and Štěpnička in [Czech. Math. J. 62 (2012), no. 3, 613-623]. The proofs rely on the clever use of the subspace theorem and the underlying ideas from the work of Corvaja and Zannier.M S C 2 0 2 0 11J68 (primary), 11J87 (secondary)

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Section: Introductionsupporting

confidence: 66%

“…Corollary 1.1 is an extension of a result due to Hančl, Kolouch, Pulcerová, and Štěpnička in [4]. It is important to note that Corvaja and Hančl [2], in 2007, proved the transcendence of infinite product using the new diophantine approximation result proved by Corvaja and Zannier [3] for the first time in the literature.…”

Section: Introductionmentioning

confidence: 55%