Values of generalized Liouville power series at algebraic numbers

TL;DR

This paper introduces generalized Liouville series and characterizes when their values at algebraic numbers are $U_m$, extending Mahler's work on lacunary power series and transcendental number approximation.

Contribution

It provides a necessary and sufficient condition for generalized Liouville series to take $U_m$-values at algebraic integers, generalizing Mahler's results.

Findings

Characterization of $U_m$-values for generalized Liouville series

Extension of Mahler's results to algebraic integers of degree $m$

Conditions under which series values are algebraic or transcendental

Abstract

For every positive integer $m$ LeVeque (1953) defined the $U_m$-numbers as the transcendental numbers that admit very good approximation by algebraic numbers of degree $m$, but not by those of smaller degree. In these terms, Mahler's $U$-numbers are the transcendental numbers which are $U_m$ for some $m$. In 1965 Mahler showed that (properly defined) lacunary power series with integers coefficients take $U$-values at algebraic numbers, unless the value is algebraic for an obvious reason. However, his argument does not specify to which $U_m$ the value belongs. In this article, we introduce the notion of generalized Liouville series, and give a necessary and sufficient condition for their values to be $U_m$. As an application, we show that a generalized Liouville series takes a $U_m$-value at a simple algebraic integer of degree $m$, unless the value is algebraic for an obvious reason. (An algebraic number $\alpha$ is called simple if the number field $\mathbb Q(\alpha)$ does not have a proper subfield other than $\mathbb Q$.)