Some inequalities for diophantine approximation by continued fractions

Abstract

Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,a2,…,an,…], let pn/qn be its nth convergent and let θn = qn|qnξ − pn|. In this paper a general method is introduced to deduce a series of inequalities involving the triple (θn−1, θn, θn+1).