Quantitative transcendence results for numbers associated with Liouville numbers

Abstract

Listen

Translate article

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