Quantitative Transcendence Results for Numbers Associated with Liouville Numbers

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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.