Quantitative transcendence results for numbers associated with Liouville numbers

Abstract:

In 1937, Franklin and Schneider generalized the Gelfond-Schneider result on the transcendence of ${\alpha ^\beta }$. They proved the following theorem: If $\beta$ is an algebraic, irrational number and $\alpha$ is "suitably well-approximated by algebraic numbers of bounded degree", then ${\alpha ^\beta }$ is transcendental. In 1964, Feldman established the algebraic independence of $\alpha$ and ${\alpha ^\beta }$ under similar conditions. We use results concerning linear forms in logarithms to give quantitative versions of the Franklin-Schneider and Feldman results.