Values of Algebraic Functions at Liouville Numbers

Source abstract

Abstract In 1953 LeVeque proved the existence of $$U_m$$ U m -numbers by showing that for some specially defined Liouville number $$\lambda$$ λ , the m th root $$\lambda ^{1/m}$$ λ 1 / m is in $$U_m$$ U m . In this article we study the following question: let u be an algebraic function of degree m and $$\lambda$$ λ a Liouville number; under which conditions is $$u(\lambda )$$ u ( λ ) a $$U_m$$ U m -number? We consider a more refined notion of $$\mathcal {L}$$ L -numbers, and show that, under very general assumptions, an algebraic function of degree m takes $$U_m$$ U m -values at all $$\mathcal {L}$$ L -numbers.