U-Numbers and T-Numbers: Some Elementary Transcendence and Algebraic Independence Results

In [2], we considered the transcendence of α^β and the algebraic independence of α and α^β when α is “suitably well-approximated by algebraic numbers of bounded degree” and β is algebraic of degree at least two. Here we consider two further cases. First, we give transcendence and algebraic independence results when both α and β are “well-approximated by algebraic numbers of bounded degree”. Second, we consider the case where α is “well-approximated by algebraic numbers of increasing degrees”. Transcendence and algebraic independence results in this setting were suggested to me by R. Tubbs who has given similar results for the Weierstrass ℘-function [8]. The second case was suggested by A. Pollington during the conference at Brigham Young. In [6], Pollington and his co-authors considered the distribution of such numbers showing that they support an M ₀-measure.