### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

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

}

*Transactions of the American Mathematical Society*, vol. 371, no. 6, pp. 3787-3804. https://doi.org/10.1090/tran/7316

**Algebraic approximations to linear combinations of powers : An extension of results by mahler and Corvaja–Zannier.** / Kulkarni, Avinash; Mavraki, Niki Myrto; Nguyen, Khoa D.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Algebraic approximations to linear combinations of powers

T2 - An extension of results by mahler and Corvaja–Zannier

AU - Kulkarni, Avinash

AU - Mavraki, Niki Myrto

AU - Nguyen, Khoa D.

PY - 2019/3/15

Y1 - 2019/3/15

N2 - 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.

AB - 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.

KW - Algebraic approximations

KW - Linear combinations of powers

KW - Linear recurrence sequences

KW - Subspace theorem

UR - http://www.scopus.com/inward/record.url?scp=85063959489&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85063959489&partnerID=8YFLogxK

U2 - 10.1090/tran/7316

DO - 10.1090/tran/7316

M3 - Article

AN - SCOPUS:85063959489

VL - 371

SP - 3787

EP - 3804

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 6

ER -