Algebraic approximations to linear combinations of powers: An extension of results by mahler and Corvaja–Zannier

Avinash Kulkarni, Niki Myrto Mavraki, Khoa D. Nguyen

Research output: Contribution to journalArticle

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 kZ < 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 languageEnglish (US)
Pages (from-to)3787-3804
Number of pages18
JournalTransactions of the American Mathematical Society
Volume371
Issue number6
DOIs
StatePublished - Mar 15 2019

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Linear Combination
Logarithmic
Approximation
Subspace Theorem
Linear Recurrence
Algebraic number
Complex number
Number field
Integer
Form

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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

In: Transactions of the American Mathematical Society, Vol. 371, No. 6, 15.03.2019, p. 3787-3804.

Research output: Contribution to journalArticle

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