On the rational approximation to linear combinations of powers

Veekesh Kumar; Gorekh Prasad

Paper

On the rational approximation to linear combinations of powers

Abstract

For a complex number

. Let

be an integer, and

be a number field. Let

be algebraic numbers with

and let

denotes the degree of

for

. Set

. In this article, we show that if the inequality

has infinitely many solutions in

with absolute logarithmic Weil height of

is small compared to

and some

, then, in particular, the tuple

is pseudo-Pisot, and at least one of

is an algebraic integer. This result can be viewed as Roth's type theorem for linear combinations of powers of algebraic numbers over

. The case

was recently proved by Kulkarni, Mavraki, and Nguyen \cite{kul}, which is a generalization of Mahler's question proved in \cite{corv}. As a consequence of our result, we obtain the following generalization of this question: let

be an algebraic number with

. For a given

, if the inequality

has infinitely many solutions in the tuples

with absolute logarithmic Weil height of

is small compared to

and

, then some power of

is a Pisot number. As an application of this result, we deduce the transcendence of certain infinite products of algebraic numbers.