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}.

On the Skolem problem and some related questions for parametric families of linear recurrence sequences

“…We also remark that Theorem 2.1 is an analogue of a result of Kulkarni, Mavraki, and Nguyen [24,Proposition 2.2], see also [4, Proposition 2.2], in which the coefficients a 1 , . .…”

“…, a k of F n pXq in (1.2) are constants rather than rational functions as in our case. Moreover, [ Furrhermore, if as in [24] the coefficients a 1 , . .…”

“…Here we study the case when both the coefficients µ i and the roots λ i are rational functions of a parameter α P Q. This scenario is close to that of Amoroso, Masser and Zannier [2], see also [24,Proposition 2.2]. In particular, we also appeal to some of the results from [2], however our method is based on a different argument.…”

On inhomogeneous extension of Thue-Roth's type inequality with moving targets

“…Recently in 2019, Kulkarni, Mavraki and Nguyen [3] generalize Mahler problem to an arbitrary linear recurrence sequence of the form {Q…”

“…For a given real number ε > 0, if there are infinitely many natural numbers n for which ||λα n + β|| < 2 −εn holds true, then α is transcendental, where ||x|| denotes the distance from its nearest integer. When α and β both are algebraic satisfying same conditions, then a particular result of Kulkarni, Mavraki and Nguyen, proved in [3] asserts that α d is a Pisot number. When β is algebraic irrational, our result implies that no algebraic number α satisfies the inequality for infinitely many natural numbers n. Also, our result strengthens a result of Wagner and Ziegler [6].…”