The conjugate property for Diophantine approximation of continued fractions

Abstract:

Let $\xi$ be an irrational number with simple continued fraction expansion $\xi = [{a_0};{a_1}, \ldots ,{a_i}, \ldots ]$, and ${p_i}/{q_i}$ be its $i$th convergent. In this paper we first prove the duality of some inequalities, and then prove the following conjugate properties for symmetric and asymmetric Diophantine approximations. (i) Among any three consecutive convergents ${p_i}/{q_i}(i = n - 1,n,n + 1)$, at least one satisfies \[ \xi - {p_i}/{q_i}| < 1/\left ( {\sqrt {a_{^{n + 1}}^2 + 4q_i^2} } \right ),\] and at least one does not satisfy this inequality. (ii) Let $\tau$ be a positive real number. Among any four consecutive convergents ${p_i}/{q_i}(i = n - 1,n,n + 1,n + 2)$, at least one satisfies \[ - 1/\left ( {\sqrt {c_{^n}^2 + 4\tau q_i^2} } \right ) < \xi - {p_i}/{q_i} < \tau /\left ( {\sqrt {c_n^2 + 4\tau q_i^2} } \right ),\] and at least one does not satisfy this inequality, where ${c_n} = {a_{n + 1}}$ if $n$ is odd, ${c_n} = {a_{n + 2}}$ if $n$ is even.