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Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Algebraic approximations to linear combinations of powers: An extension of results by Mahler and Corvaja–Zannier
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by Avinash Kulkarni, Niki Myrto Mavraki and Khoa D. Nguyen PDF
Trans. Amer. Math. Soc. 371 (2019), 3787-3804 Request permission

Abstract:

For every complex number $x$, let $\Vert x\Vert _{\mathbb {Z}}:=\min \{|x-m|:\ m\in \mathbb {Z}\}$. Let $K$ be a number field, let $k\in \mathbb {N}$, and let $\alpha _1,\ldots ,\alpha _k$ be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of $\theta \in (0,1)$ such that there are infinitely many tuples $(n,q_1,\ldots ,q_k)$ satisfying $\Vert q_1\alpha _1^n+\cdots +q_k\alpha _k^n\Vert _{\mathbb {Z}}<\theta ^n$ where $n\in \mathbb {N}$ and $q_1,\ldots ,q_k\in K^*$ have small logarithmic height compared to $n$. In the special case when $q_1,\ldots ,q_k$ have the form $q_i=qc_i$ for fixed $c_1,\ldots ,c_k$, our work yields results on algebraic approximations of $c_1\alpha _1^n+\cdots +c_k\alpha _k^n$ of the form $\frac {m}{q}$ with $m\in \mathbb {Z}$ and $q\in K^*$ (where $q$ has small logarithmic height compared to $n$). Various results on linear recurrence sequences also follow as an immediate consequence. The case where $k=1$ and $q_1$ is essentially a rational integer was obtained by Corvaja and Zannier and settled a long-standing question of Mahler. The use of the Subspace Theorem based on work of Corvaja–Zannier, together with several modifications, plays an important role in the proof of our results.
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Additional Information
  • Avinash Kulkarni
  • Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada
  • Email: akulkarn@sfu.ca
  • Niki Myrto Mavraki
  • Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
  • Address at time of publication: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
  • MR Author ID: 1050391
  • Email: myrtomav@northwestern.edu
  • Khoa D. Nguyen
  • Affiliation: Department of Mathematics, University of British Columbia — and — Pacific Institute for The Mathematical Sciences, Vancouver, British Columbia V6T 1Z2, Canada
  • Address at time of publication: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, Canada
  • MR Author ID: 886774
  • Email: dangkhoa.nguyen@ucalgary.ca
  • Received by editor(s): February 25, 2017
  • Received by editor(s) in revised form: May 24, 2017, and June 20, 2017
  • Published electronically: November 16, 2018
  • Additional Notes: The first author was partially supported by NSERC
    The third author was partially supported by a UBC-PIMS fellowship
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 371 (2019), 3787-3804
  • MSC (2010): Primary 11J68, 11J87; Secondary 11B37, 11R06
  • DOI: https://doi.org/10.1090/tran/7316
  • MathSciNet review: 3917208