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- W2962907105 abstract "For every complex number $x$, let $Vert xVert _{mathbb {Z}}:=min {|x-m|: min mathbb {Z}}$. Let $K$ be a number field, let $kin mathbb {N}$, 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 $Vert q_1alpha _1^n+cdots +q_kalpha _k^nVert _{mathbb {Z}}<theta ^n$ where $nin mathbb {N}$ and $q_1,ldots ,q_kin K^*$ 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 $c_1alpha _1^n+cdots +c_kalpha _k^n$ of the form $frac {m}{q}$ with $min mathbb {Z}$ and $qin K^*$ (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 CorvajaâZannier, together with several modifications, plays an important role in the proof of our results." @default.
- W2962907105 created "2019-07-30" @default.
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- W2962907105 date "2018-11-16" @default.
- W2962907105 modified "2023-09-25" @default.
- W2962907105 title "Algebraic approximations to linear combinations of powers: An extension of results by Mahler and Corvaja–Zannier" @default.
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- W2962907105 doi "https://doi.org/10.1090/tran/7316" @default.
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