A note on some applications of the subspace theorem

Abstract

Let \(f(z) = \sum _{n=0}^{\infty } a_n z^n\) be a power series with integer coefficients and converging in the disc \(D = \{ z : |z| < R \}\) for some \(R > 0\). In 1985, Laohakosol proved, using Ridout theorem, that the largest prime factors of partial sums of f(b) for a rational number \(0< |b| < R\) is unbounded, if f(b) is a non-zero algebraic number. In this article, we prove, using the subspace theorem, similar results for other approximation of f(b). Moreover, we prove the number field analogue of Laohakosol’s result.

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Acknowledgements

We would like to thank Prof. R. Thangadurai for his valuable suggestions and comments. The authors are grateful to anonymous referee for his/her valuable suggestions and remarks which improved the exposition of the paper. We would like to acknowledge the Department of Atomic Energy, Govt. of India for providing the research grant.

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Correspondence to Debasish Karmakar.

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Agnihotri, R., Karmakar, D. & Kumar, V. A note on some applications of the subspace theorem. Res. number theory 7, 14 (2021). https://doi.org/10.1007/s40993-021-00238-0

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Keywords

  • Schmidt subspace theorem
  • Rational approximation

Mathematics Subject Classification

  • Primary 11J87
  • Secondary 11J68