### Abstract

For every complex number x, let‖x‖
_{Z}
:= min{|x − m|: m ∈ Z}. Let K be a number field, let k ∈ N, andletα
_{1}
,…,α
_{k}
be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of θ ∈ (0, 1) such that there are infinitely many tuples (n, q
_{1}
,…,q
_{k}
) satisfying ‖q
_{1}
α
^{n}
_{1}
+ ··· + q
_{k}
α
^{n}
_{k}
‖
_{Z}
< θ
^{n}
where n ∈ N and q
_{1}
,…,q
_{k}
∈ K
^{∗}
have small logarithmic height compared to n. In the special case when q
_{1}
,…,q
_{k}
have the form q
_{i}
= qc
_{i}
for fixed c
_{1}
,…,c
_{k}
, our work yields results on algebraic approximations of c
_{1}
α
^{n}
_{1}
+ ···+ c
_{k}
α
^{n}
_{k}
of the form
^{m}
_{q}
with m ∈ Z and q ∈ 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 =1andq
_{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.

Original language | English (US) |
---|---|

Pages (from-to) | 3787-3804 |

Number of pages | 18 |

Journal | Transactions of the American Mathematical Society |

Volume | 371 |

Issue number | 6 |

DOIs | |

State | Published - Mar 15 2019 |

### Keywords

- Algebraic approximations
- Linear combinations of powers
- Linear recurrence sequences
- Subspace theorem

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Transactions of the American Mathematical Society*,

*371*(6), 3787-3804. https://doi.org/10.1090/tran/7316