The Diophantine Approximation of Roots of Positive Integers

Rece nt ly Schinzel [4] I and Dave nport [1] have each obtain ed a res u lt of the followin g so rt : Let a be a real algeb raic numbe r of degree r ~ 2. Le t s be a positive real numbe r large r than s( r), whe re for Daven port r > s( r ) = tr+ 0(1 ) > t r while for Schinzel s( r) = 3(r/2) 1/2 Th e n th ere ex is ts an e ffectively co mputable positive integer qo(a , s) s uc h th at , with a t mos t one exce p· tion, every pair of relatively prim e integers p and q with q ~ qo (a) sa tis fi es th e in equ ality